Sloane's Database of Integer Sequences, Part 27 Part of the On-Line Encyclopedia of Integer Sequences This is a section of the main database for the On-Line Encyclopedia of Integer Sequences. For more information see the following pages: ( www.research.att.com/~njas/sequences/ then ) Seis.html: Welcome index.html: Lookup indexfr.html: Francais demo1.html: Demos Sindx.html: Index WebCam.html: WebCam Submit.html: Contribute new sequence or comment eishelp1.html: Internal format eishelp2.html: Beautified format transforms.html: Transforms Spuzzle.html: Puzzles Shot.html: Hot classic.html: Classics ol.html: Superseeker JIS/index.html: Journal of Integer Sequences pages.html: More pages Maintained by: N. J. A. Sloane (njas@research.att.com), home page: www.research.att.com/~njas/ (start) %I A063915 %S A063915 1,3,5,9,13,17,21,29,37,45,53,61,69,77,85,101,117,133,149,165,181, %T A063915 197,213,229,245,261,277,293,309,325,341,373,405,437,469,501,533, %U A063915 565,597,629,661,693,725,757,789,821,853,885,917,949,981,1013,1045 %N A063915 G.f.: (1 + Sum_{ i = 0..infinity} 2^i*x^(2^(i+1)-1)) / (1-x)^2. %K A063915 nonn %O A063915 0,2 %A A063915 njas, Sep 01 2001 %I A049690 %S A049690 0,1,3,5,9,13,17,23,31,37,45,55,63,75,87,95,111,127,139,157,173 %N A049690 a(n)=b(2n), where b=A049689. %K A049690 nonn %O A049690 0,3 %A A049690 Clark Kimberling, ck6@cedar.evansville.edu %I A061571 %S A061571 1,3,5,9,13,17,23,31,37,45,55,63,75,87,95,111,127,139,157,173,185,205, %T A061571 227,243,263,287,305,329,357,373,403,435,455,487,511,535,571,607,631, %U A061571 663,703,727,769,809,833,877,923,955,997,1037,1069,1117,1169,1205,1245 %N A061571 a(n)=Sum_{k=1..n} phi(2*k), where phi = Euler totient function, cf. A000010. %C A061571 May be the same as A049690. %K A061571 nonn %O A061571 1,2 %A A061571 Vladeta Jovovic (vladeta@Eunet.yu), May 18 2001 %I A007664 M2449 %S A007664 1,3,5,9,13,17,25,33,41,49,65,81,97,113,129,161,193,225,257,289, %T A007664 321,385,449,513,577,641,705,769,897,1025,1153,1281,1409,1537,1665, %U A007664 1793,2049,2305,2561,2817,3073,3329,3585,3841,4097,4609,5121,5633 %N A007664 Reve's puzzle: Tower of Hanoi with 4 pegs. %C A007664 The formula given below involving A002024 is conjectured to give the optimal solution for all n. %D A007664 J.-P. Allouche, Note on the cyclic towers of Hanoi, Theoret. Comput. Sci., 123 (1994), 3-7. %D A007664 A. Brousseau, Tower of Hanoi with more pegs, J. Recreational Math., 8 (1972), 169-176. %D A007664 D. Wood, Towers of Brahma and Hanoi revisited, J. Recreational Math., 14 (1981), 17-24. %H A007664 Towers of Hanoi %F A007664 A007664(n) = 1 + (n + A003056(n) - 1 - A003056(n)*(A003056(n) + 1)/2)*2^A003056(n) - Daniele Parisse (daniele.parisse@m.dasa.de), Feb 06 2001 %F A007664 Sum(i*2^(i-1),i=1..n)-(A002024(n)*(A002024(n)+1)/2-n)*2^(A002024(n)-1). %p A007664 A007664:=n->add(i*2^(i-1),i=1..n)-(A002024(n)*(A002024(n)+1)/2-n)*2^(A002024(n)-1); %Y A007664 Cf. A007665, A003056. %K A007664 nonn,nice %O A007664 1,2 %A A007664 njas, mb, Robert G. Wilson v (rgwv@kspaint.com) %I A032635 %S A032635 0,0,1,3,5,9,13,18,23,29,36,44,52,62,72,82,94,106,119,132,147,162, %T A032635 178,194,211,229,248,268,288,309,331,353,376,400,425,450,476,503,531, %U A032635 559,588,618,648,680,712,744,778,812,847,883,919,956,994,1033,1072 %N A032635 [ n^2 / e ]. %K A032635 nonn,easy %O A032635 0,4 %A A032635 Patrick De Geest (pdg@worldofnumbers.com), may 1998. %I A036713 %S A036713 0,1,3,5,9,13,18,23,29,38,45,54,63,74,85,96,110,124,138,153,167,185, %T A036713 203,221,239,260,281,301,324,346,371,395,419,445,473,501,530,558,589, %U A036713 619,652,683,716,751,784,820,855,893,931,970,1010 %N A036713 a(n)=number of Gaussian integers z=a+bi satisfying |z|<=n+1/2, a>=0, 0<=b=2q-1. Terms were calculated by Weissman, Carmody and McCranie. %e A058989 The 4th prime is 7. Nine is the maximum number of consecutive integers such that each is disibile by 2, 3, 5, or 7. (Example: 2 through 10) So a(4)=9. %Y A058989 Cf. A000040. %K A058989 nice,nonn %O A058989 1,2 %A A058989 Jud McCranie (jud.mccranie@mindspring.com), Jan 16 2001 %I A049691 %S A049691 0,3,5,9,13,21,25,37,45,57,65,85,93,117,129,145,161,193,205,241,257, %T A049691 281,301,345,361,401,425,461,485,541,557,617,649,689,721,769,793,865, %U A049691 901,949,981 %N A049691 a(n)=T(n,n), array T as in A049687. Also a(n)=T(2n,2n), array T given by A049639. %K A049691 nonn %O A049691 0,2 %A A049691 Clark Kimberling, ck6@cedar.evansville.edu %I A052282 %S A052282 1,1,3,5,9,13,22,30,45,61,85,111,149,189,244,304,381,465,571,685,825, %T A052282 977,1158,1354,1585,1833 %N A052282 3 x3 stochastic matrices under row and column permutations. %e A052282 There are 5 nonisomorphic 3x3 matrices with row and column sums 3: %e A052282 [0 0 3] [0 0 3] [0 1 2] [0 1 2] [1 1 1] %e A052282 [0 3 0] [1 2 0] [1 1 1] [1 2 0] [1 1 1] %e A052282 [3 0 0] [2 1 0] [2 1 0] [2 0 1] [1 1 1] %Y A052282 Cf. A002817, A052280, A052281. May be same as A001993? %K A052282 more,nonn %O A052282 0,3 %A A052282 Vladeta Jovovic (vladeta@Eunet.yu), Feb 06 2000 %I A001993 M2452 N0973 %S A001993 1,1,3,5,9,13,22,30,45,61,85,111,150,190,247,309,390,478,593,715,870, %T A001993 1038,1243,1465,1735,2023,2368,2740,3175,3643,4189,4771,5443,6163,6982, %U A001993 7858,8852,9908,11098,12366,13780,15284,16958,18730,20692,22772,25058 %N A001993 Expansion of a generating function. %D A001993 A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419. %F A001993 G.f.: 1/((1-x^2)*(1-x^4)^2*(1-x^6)^2*(1-x^8)) %K A001993 nonn,easy %O A001993 0,3 %A A001993 njas %E A001993 More terms from James A. Sellers (sellersj@math.psu.edu), Feb 09 2000 %I A054066 %S A054066 1,3,5,9,14,19,26,33,42,52,62,74,87,100,115,130,147,165,183,203,223, %T A054066 245,268,291,316,342,368,396,424,454,485,516,549,583,617,653,689,727, %U A054066 766,805,846,887,930,974,1018,1064,1111,1158,1207 %N A054066 Position of n-th 1 in A054065. %K A054066 nonn %O A054066 1,2 %A A054066 Clark Kimberling, ck6@cedar.evansville.edu %I A053618 %S A053618 0,0,0,1,1,3,5,9,14,21,30,42,55,72,91,114,140,170,204,243,285,333, %T A053618 385,443,506,575,650,732,819,914,1015,1124,1240,1364,1496,1637, %U A053618 1785,1943,2109,2285,2470,2665,2870,3086,3311,3548,3795,4054,4324 %N A053618 Ceiling(C(n,4)/n). %D A053618 R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 26 (1980), 37-43. %Y A053618 Cf. A007997, A008646, A053643, A032192, A053731, A053733. %K A053618 nonn %O A053618 1,6 %A A053618 njas, Mar 25 2000 %I A032801 %S A032801 0,0,0,0,1,3,5,9,14,22,30,42,55,73,91,115,140,172,204,244, %T A032801 285,335,385,445,506,578,650,734,819,917,1015,1127,1240, %U A032801 1368,1496,1640,1785,1947,2109,2289,2470,2670,2870,3090 %N A032801 Number of unordered sets a,b,c,d of distinct integers from 1..n such that a+b+c+d=0 (mod n). %D A032801 Barnes, Acta Arith., 5 (1958), p. 65. %p A032801 f:=n-> if n mod 2 <> 0 then (n-1)*(n-2)*(n-3)/24 elif n mod 4 = 0 then (n-4)*(n^2-2*n+6)/24 else (n-2)*(n^2-4*n+6)/24; fi; %K A032801 nonn %O A032801 0,6 %A A032801 njas %I A033818 %S A033818 3,5,9,14,22,34,53,83,131,208,332,532,855,1377,2221,3586,5794,9366, %T A033818 15145,24495,39623,64100,103704,167784,271467,439229,710673,1149878, %U A033818 1860526,3010378 %N A033818 Convolution of natural numbers n >= 1 with Lucas numbers L(k) for k >= -2. %F A033818 a(n) = L(1)*(F(n+1)-1)+L(0)*F(n)-L(-1)*n, F(n): Fibonacci (A000045), L(n): Lucas (A000032) with L(-n)=(-1)^n*L(n) %F A033818 G.F. x*(3-4*x)/((1-x-x^2)*(1-x)^2), 3=L(-2), -4=+L(-3). %Y A033818 Cf. A023548, A023537, A033811. %K A033818 easy,nonn %O A033818 1,1 %A A033818 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de) %I A061556 %S A061556 1,3,5,9,14,23,43,79,149,263,461,823,1451,2549,4483 %N A061556 Smallest number m such that the abundancy-index of m! exceeds n exceeds n. %C A061556 Floor[Sigma[m! ]/m! ] = n; note that abundancy-index [= sigma(u)/u] here is not necessarily an integer. %H A061556 Achim Flammenkamp, The Multiply Perfect Numbers Page %H A061556 Fred Helenius, Link to Glossary and Lists %F A061556 a(n)=Min{w | Floor[Sigma(w!)/w!=n] %e A061556 Floor[ sigma(842!)/842! ]=11 while Floor[ sigma(843!)/843! ]=12 %Y A061556 Cf. A000142, A000203. %K A061556 more,nonn %O A061556 0,2 %A A061556 Labos E. (labos@ana1.sote.hu), May 17 2001 %I A053993 %S A053993 1,1,3,5,9,14,24,35,55,81,120,171,248,345,486,669,920,1246,1690,2256, %T A053993 3014,3984,5253,6870,8970,11618,15022,19306,24745,31557,40154,50845, %U A053993 64244,80850,101501,126982,158514,197218,244865,303143,374497,461435 %N A053993 The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row. %D A053993 Andrews, George E., Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984. %F A053993 Generating function in Andrews' Memoir %K A053993 easy,nonn %O A053993 1,3 %A A053993 James A. Sellers (sellersj@math.psu.edu), Apr 04 2000 %I A018634 %S A018634 1,3,5,9,15,17,45,51,85,153,255,765 %N A018634 Divisors of 765. %K A018634 nonn,fini,full %O A018634 0,2 %A A018634 njas %I A029533 %S A029533 1,3,5,9,15,19,21,43,57,255,345,385,505,1131,1395,1585,7205,11565, %T A029533 38949 %N A029533 n divides the (left) concatenation of all numbers <= n written in base 16 (most significant digit on right). %C A029533 This sequence differs from A061969 in that all least significant zeros are removed before concatenation. %e A029533 See A029519 for example. %Y A029533 Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978. %K A029533 nonn,base,more %O A029533 1,2 %A A029533 Olivier Gerard (ogerard@ext.jussieu.fr) %E A029533 Additional comments and more terms from Larry Reeves (larryr@acm.org), Jun 04 2001 %I A018685 %S A018685 1,3,5,9,15,19,45,57,95,171,285,855 %N A018685 Divisors of 855. %K A018685 nonn,fini,full %O A018685 0,2 %A A018685 njas %I A014957 %S A014957 1,3,5,9,15,21,25,27,39,45,55,63,75,81,105,117,125,135,147,155,165, %T A014957 171,189,195,205,225,243,273,275,315,333,351,375,405,441,465,495,507, %U A014957 513,525,567,585,605,609,615,625,657,675,729,735,775,819,825,855,903 %N A014957 n divides s(n), where s(1)=1, s(n)=s(n-1)+n*16^(n-1). %K A014957 nonn %O A014957 1,2 %A A014957 Olivier Gerard (ogerard@ext.jussieu.fr) %I A014876 %S A014876 1,3,5,9,15,21,25,27,45,55,63,75,81,93,105,135,147,165,171,189,225, %T A014876 243,275,279,315,355,405,441,465,495,513,525,567,605,609,651,675,729, %U A014876 735,825,837,855,903,915,945,1029,1065,1155,1197,1215,1265,1323,1395 %N A014876 n divides s(n), where s(1)=1, s(n)=25*s(n-1)+n. %K A014876 nonn %O A014876 1,2 %A A014876 njas, Olivier Gerard (ogerard@ext.jussieu.fr) %I A045602 %S A045602 1,3,5,9,15,21,25,27,45,55,63,75,81,105,125,135,147,165,189,225,243, %T A045602 275,315,375,405,441,465,495,525,567,605,625,675,729,735,825,945,1029, %U A045602 1125,1155,1215,1323,1375,1395,1485,1575,1701,1815,1875,2025,2187,2205 %N A045602 n | 9^n + 6^n. %K A045602 nonn %O A045602 1,2 %A A045602 dww %I A029470 %S A029470 1,3,5,9,15,21,25,35,64,75,125,192,320,345,375,625,675,825,875,945,960, %T A029470 1117,1125,1155,1255,1344,1375,1485,1575,1600,1728,1875,1925,2240,2475, %U A029470 2625,2880,3125,3375,3465,3520,4032,4065,4125,4375,4725,4800,5625,5775 %N A029470 n divides the (right) concatenation of all numbers <= n written in base 25 (most significant digit on left). %Y A029470 Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978. %K A029470 nonn,base %O A029470 1,2 %A A029470 Olivier Gerard (ogerard@ext.jussieu.fr) %E A029470 More terms from Larry Reeves (larryr@acm.org), May 02 2001 %I A029518 %S A029518 1,3,5,9,15,21,39,83,96,288,303,864,1824,2421,2496,2592,2817,3328,6299, %T A029518 9440,13632,18592,26049,64857,69696,71904,79872,94848,120384,258111, %U A029518 287232 %N A029518 n divides the (right) concatenation of all numbers <= n written in base 25 (most significant digit on right). %C A029518 This sequence differs from A061954 in that all least significant zeros are kept during concatenation. %e A029518 See A029495 for example. %Y A029518 Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978. %K A029518 nonn,base %O A029518 1,2 %A A029518 Olivier Gerard (ogerard@ext.jussieu.fr) %E A029518 Additional comments and more terms from Larry Reeves (larryr@acm.org), May 25 2001 %I A061954 %S A061954 1,3,5,9,15,21,96,99,259,1317,2112 %N A061954 n divides the (right) concatenation of all numbers <= n written in base 25 (most significant digit on right). %C A061954 This sequence differs from A029518 in that all least significant zeros are removed before concatenation. %e A061954 See A061931 for example. %Y A061954 Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978. %K A061954 nonn,base,more %O A061954 1,2 %A A061954 Larry Reeves (larryr@acm.org), May 24 2001 %I A022940 %S A022940 1,3,5,9,15,22,30,40,51,63,76,90,106,123,141,160,180,201,224,248, %T A022940 273,299,326,354,383,414,446,479,513,548,584,621,659,698,739,781, %U A022940 824,868,913,959,1006,1054,1103,1153,1205,1258,1312,1367,1423,1480 %N A022940 a(n) = a(n-1) + c(n-2) for n >= 3, a( ) increasing, given a(1)=1, a(2)=3; where c( ) is complement of a( ). %K A022940 nonn %O A022940 1,2 %A A022940 Clark Kimberling (ck6@cedar.evansville.edu) %I A025207 %S A025207 0,0,1,3,5,9,15,22,31,42,55,71,88,108,131,157,185,216,249,286,326,369,415, %T A025207 464,516,572,632,694,760,830,903,980,1060,1144,1232,1323,1418,1517,1620,1727, %U A025207 1837,1952,2070,2192,2319,2449,2584,2722,2865,3011,3162,3317,3476,3639,3807 %N A025207 a(n) = [ Sum{(log(j)-log(i))^3} ], 2 <= i < j <= n. %K A025207 nonn %O A025207 3,4 %A A025207 Clark Kimberling (ck6@cedar.evansville.edu) %I A027688 %S A027688 3,5,9,15,23,33,45,59,75,93,113,135,159,185,213,243,275,309,345,383, %T A027688 423,465,509,555,603,653,705,759,815,873,933,995,1059,1125,1193, %U A027688 1263,1335,1409,1485,1563,1643,1725,1809,1895,1983,2073,2165,2259 %N A027688 Numbers of form n^2 + (n+3). %H A027688 P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X) %K A027688 nonn %O A027688 0,1 %A A027688 Patrick De Geest (pdg@worldofnumbers.com) %I A033498 %S A033498 1,3,5,9,15,23,35,53,73,101,141,185,247,329,417,533,689, %T A033498 853,1061,1331,1613,1977,2429,2899,3485,4227,4989,5915, %U A033498 7049,8211,9643,11357,13111,15229,17799,20413,23497,27167 %N A033498 See program line. %p A033498 A033498:=proc(n) option remember; if n <= 1 then 1 else A033498(n-1)+A033498(round(2*(n-1)/3))+A033498(round((n-1)/3)); fi; end; %K A033498 nonn %O A033498 1,2 %A A033498 njas %I A057259 %S A057259 1,3,5,9,15,25,27,39,45,75,81,117,125,135,195,225,243,297,351,375,405, %T A057259 507,585,625,645,675,729,975,1053,1125,1215,1287,1521,1755,1875,2025, %U A057259 2187,2277,2535,2925,3125,3159,3375,3645,4257,4563,4875,5085,5265,5577 %N A057259 n | 10^n + 9^n + 8^n + 7^n + 6^n + 5^n. %t A057259 Select[ Range[ 10^4 ], Mod[ PowerMod[ 10, #, # ] + PowerMod[ 9, #, # ] + PowerMod[ 8, #, # ] + PowerMod[ 7, #, # ] + PowerMod[ 6, #, # ] + PowerMod[ 5, #, # ], # ] == 0 & ] %K A057259 nonn %O A057259 1,2 %A A057259 Robert G. Wilson v (rgwv@kspaint.com), Sep 21 2000 %I A045599 %S A045599 1,3,5,9,15,25,27,45,55,75,81,125,135,165,171,225,243,275,375,405,495, %T A045599 513,605,625,675,729,825,855,1125,1215,1375,1485,1539,1815,1875,2025, %U A045599 2187,2475,2565,3025,3125,3249,3375,3645,4125,4275,4455,4617,4965,5445 %N A045599 n | 10^n + 5^n. %K A045599 nonn %O A045599 1,2 %A A045599 dww %I A045604 %S A045604 1,3,5,9,15,25,27,45,57,75,81,125,135,171,225,243,285,375,405,513,625, %T A045604 675,729,855,915,1083,1125,1215,1425,1539,1875,2025,2187,2525,2565, %U A045604 2745,3125,3249,3305,3375,3645,4275,4575,4617,5415,5625,6075,6561,7125 %N A045604 n | 8^n + 7^n. %K A045604 nonn %O A045604 1,2 %A A045604 dww %I A057251 %S A057251 1,3,5,9,15,25,27,45,75,81,87,125,135,155,225,243,265,297,375,405,625, %T A057251 675,729,775,1107,1125,1215,1875,2025,2187,3125,3375,3645,3875,4779, %U A057251 4805,5625,6075,6561,9369,9375,10125,10935,12175,12879,15625,16875 %N A057251 n | 11^n + 10^n + 9^n + 8^n + 7^n. %t A057251 Select[ Range[ 10^5 ], Mod[ PowerMod[ 11, #, # ] + PowerMod[ 10, #, # ] + PowerMod[ 9, #, # ] + PowerMod[ 8, #, # ] + PowerMod[ 7, #, # ], # ] == 0 & ] %K A057251 nonn %O A057251 1,2 %A A057251 Robert G. Wilson v (rgwv@kspaint.com), Sep 21 2000 %I A015965 %S A015965 1,3,5,9,15,25,27,45,75,81,125,135,171,183,225,243,355,375,405,505, %T A015965 513,549,625,675,729,855,915,1065,1125,1215,1515,1539,1647,1775, %U A015965 1875,2025,2187,2525,2565,2745,3125,3195,3249,3375,3645,4275,4545 %N A015965 n | 14^n + 1. %K A015965 nonn %O A015965 0,2 %A A015965 Robert G. Wilson v (rgwv@kspaint.com) %I A056741 %S A056741 1,3,5,9,15,25,27,45,75,81,125,135,225,243,261,295,375,405,625,675,729, %T A056741 925,1125,1215,1875,2025,2187,3125,3375,3645,4077,4833,5139,5625,6075, %U A056741 6345,6561,9375,10125,10935,15625,16875,17895,18125,18225,18495,19683 %N A056741 n | 5^n + 4^n + 3^n + 2^n + 1^n. %t A056741 Do[ If[ Mod[ PowerMod[ 5, n, n ] + PowerMod[ 4, n, n ] + PowerMod[ 3, n, n ] + PowerMod[ 2, n, n ] + 1, n ] == 0, Print[ n ] ], {n, 1, 10^6} ] %Y A056741 Cf. A001552. %K A056741 nonn %O A056741 1,2 %A A056741 Robert G. Wilson v (RGWv@kspaint.com), Aug 25 2000 %I A057235 %S A057235 1,3,5,9,15,25,27,45,75,81,125,135,225,243,265,375,405,625,675,729,765, %T A057235 1005,1125,1143,1215,1875,2025,2125,2151,2187,2403,3125,3375,3645,5427, %U A057235 5625,6075,6561,9375,10125,10935,13095,15625,16875,17145,18225,19683 %N A057235 n | 6^n + 5^n + 4^n. %t A057235 Select[ Range[ 10^6 ], Mod[ PowerMod[ 6, #, # ] + PowerMod[ 5, #, # ] + PowerMod[ 4, #, # ], # ] == 0 & ] %K A057235 nonn %O A057235 1,2 %A A057235 Robert G. Wilson v (RGWv@kspaint.com), Sep 20 2000 %I A057289 %S A057289 1,3,5,9,15,25,27,45,75,81,125,135,225,243,345,375,405,625,675,729, %T A057289 1025,1125,1215,1863,1875,2025,2187,2375,2875,3125,3375,3645,4395,5625, %U A057289 5875,6075,6561,8145,9225,9375,10125,10935,14015,14523,15525,15625 %N A057289 n | 12^n + 11^n + 10^n + 9^n + 8^n + 7^n + 6^n + 5^n + 4^n + 3^n. %t A057289 Select[ Range[ 10^5 ], Mod[ PowerMod[ 12, #, # ] + PowerMod[ 11, #, # ] + PowerMod[ 10, #, # ] + PowerMod[ 9, #, # ] + PowerMod[ 8, #, # ] + PowerMod[ 7, #, # ] + PowerMod[ 6, #, # ] + PowerMod[ 5, #, # ] + PowerMod[ 4, #, # ] + PowerMod[ 3, #, # ], # ] == 0 & ] %K A057289 nonn %O A057289 1,2 %A A057289 Robert G. Wilson v (rgwv@kspaint.com), Sep 22 2000 %I A056754 %S A056754 1,3,5,9,15,25,27,45,75,81,125,135,225,243,375,405,423,625,675,729,765, %T A056754 801,895,1017,1125,1175,1215,1875,2025,2125,2187,3125,3375,3645,4755, %U A056754 5625,6075,6561,9375,9585,10125,10935,15625,16875,17925,18225,19683 %N A056754 n | 9^n + 8^n + 7^n + 6^n + 5^n + 4^n + 3^n + 2^n + 1^n. %t A056754 Do[ If[ Mod[ PowerMod[ 9, n, n ] + PowerMod[ 8, n, n ] + PowerMod[ 7, n, n ] + PowerMod[ 6, n, n ] + PowerMod[ 5, n, n ] + PowerMod[ 4, n, n ] + PowerMod[ 3, n, n ] + PowerMod[ 2, n, n ] + 1, n ] == 0, Print[ n ] ], {n, 1, 10^6} ] %Y A056754 Cf. A001556. %K A056754 nonn %O A056754 1,2 %A A056754 Robert G. Wilson v (RGWv@kspaint.com), Aug 25 2000 %I A003593 %S A003593 1,3,5,9,15,25,27,45,75,81,125,135,225,243,375,405,625,675,729,1125, %T A003593 1215,1875,2025,2187,3125,3375,3645,5625,6075,6561,9375,10125,10935, %U A003593 15625,16875,18225,19683,28125,30375,32805,46875,50625,54675,59049 %N A003593 Of form 3^i*5^j. %K A003593 nonn %O A003593 0,2 %A A003593 njas %I A018586 %S A018586 1,3,5,9,15,25,27,45,75,135,225,675 %N A018586 Divisors of 675. %K A018586 nonn,fini,full %O A018586 0,2 %A A018586 njas %I A029877 %S A029877 1,3,5,9,15,25,38,61,91,137,200,293,416,595,832,1161,1601, %T A029877 2198,2983,4042,5419,7245,9615,12714,16703,21879,28492, %U A029877 36989,47795,61561,78956,100986,128653,163461,206981,261393 %N A029877 Euler transform of Thue-Morse sequence A001285. %H A029877 N. J. A. Sloane, Transforms %K A029877 nonn %O A029877 0,2 %A A029877 njas %I A053523 %S A053523 1,1,3,5,9,15,25,41,67,109,177,287,465,753,305,265,571,837,705,193,899, %T A053523 547,181,729,911,821,867,845,857,213,67,281,349,631,981,807,895,213, %U A053523 555,769,663,717,691,705,699,703,351,33,385,419,805,613 %N A053523 a(n)=[A*a(n-1)+B*a(n-2)+C]/p^r, where p^r is the highest power of p dividing [A*a(n-1)+B*a(n-2)+C], A=1.0001, B=1.0001, C=1.9, p=2. %C A053523 Becomes a cyclic sequence whose period is 269. %Y A053523 A053521, A053522. %K A053523 nonn %O A053523 1,3 %A A053523 Yasutoshi Kohomoto (kohmoto@z2.zzz.or.jp) %I A053522 %S A053522 1,1,3,5,9,15,25,41,67,109,177,287,465,753,1219,1973,3193,323,3517, %T A053522 3841,115,3957,4073,251,4325,4577,1113,1423,2537,3961,1625,1397,3023, %U A053522 4421,3723,4073,3899,3987,493,4481,4975,4729,4853,599,2727,3327,757 %N A053522 a(n)=[A*a(n-1)+B*a(n-2)+C]/p^r, where p^r is the highest power of p dividing [A*a(n-1)+B*a(n-2)+C], A=1.0001, B=1.0001, C=1.5, p=2. %Y A053522 A053521, A053523, A028948. %K A053522 nonn %O A053522 1,3 %A A053522 Yasutoshi Kohomoto (kohmoto@z2.zzz.or.jp) %E A053522 At 10087-th term, it becomes a cyclic sequence whose period is 10086. %I A053521 %S A053521 1,1,3,5,9,15,25,41,67,109,177,287,465,753,1219,1973,3193,5167,8361, %T A053521 6765,1891,8657,5275,6967,3061,5015,8077,6547,7313,6931,7123,1757,881, %U A053521 665,9547,5107,229,5337,5567,5453,5511,5483,2749,8233,1373,9607 %N A053521 a(n)=[A*a(n-1)+B*a(n-2)+C]/p^r, where p^r is the highest power of p dividing [A*a(n-1)+B*a(n-2)+C], A=1.0001, B=1.0001, C=1, p=2. %Y A053521 Cf. A053522, A053523, A028948. %K A053521 nonn %O A053521 1,3 %A A053521 Yasutoshi Kohmoto (kohmoto@z2.zzz.or.jp) %E A053521 Becomes a cyclic sequence whose period is 4793. If A=1, B=1, C=0, p=1, a(1)=a(2)=1 then it is the Fibonacci Sequence. %I A001595 M2453 N0974 %S A001595 1,1,3,5,9,15,25,41,67,109,177,287,465,753,1219,1973,3193, %T A001595 5167,8361,13529,21891,35421,57313,92735,150049,242785,392835, %U A001595 635621,1028457,1664079,2692537,4356617,7049155,11405773 %N A001595 a(n) = a(n-1) + a(n-2) + 1. %C A001595 2-ranks of difference sets constructed from Segre hyperovals. %D A001595 R. Evans, H. D. L. Hollmann, C. Krattenthaler, Q. Xiang, Gauss Sums, Jacobi Sums, and p-Ranks of Cyclic Difference Sets, J. Combin. Theory Ser. A 87 (1999), 74-119. %D A001595 D. Singmaster, Some counterexamples and problems on linear recurrences, Fib. Quart. 8 (1970), 264-267. %D A001595 Q. Xiang, On Balanced Binary Sequences with Two-Level Autocorrelation Functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156. %H A001595 Supplement to "Gauss Sums, Jacobi Sums, and p-Ranks ..." %H A001595 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1019 %F A001595 G.f.: (1+x-x^2)/(1-2*x+x^3). a(n) = 2/sqrt(5)*((1+sqrt(5))/2)^(n+1) - 2/sqrt(5)*((1-sqrt(5))/2)^(n+1) - 1. %p A001595 L:=1,3: for i from 3 to 100 do l:=nops([ L ]): L:=L,op(l,[ L ])+op(l-1,[ L ])+1: od: [ L ]; %t A001595 Join[ {1,3},Table[ a[ 1 ]=1;a[ 2 ]=3;a[ i ]=a[ i-1 ]+a[ i-2 ]+1,{i,3,100} ] ] %Y A001595 Cf. A049112, A049114. %K A001595 nonn,easy,nice %O A001595 0,3 %A A001595 njas %E A001595 Additional comments from Christian Krattenthaler (kratt@ap.univie.ac.at). %I A061969 %S A061969 1,3,5,9,15,25,41,1149,1755,2009,2815,6981,19117 %N A061969 n divides the (left) concatenation of all numbers <= n written in base 16 (most significant digit on right). %C A061969 This sequence differs from A029533 in that all least significant zeros are kept during concatenation. %e A061969 See A061955 for example. %Y A061969 Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978. %K A061969 nonn,base,more %O A061969 1,2 %A A061969 Larry Reeves (larryr@acm.org), May 24 2001 %I A034084 %S A034084 3,5,9,15,25,42,70,119,202,343,583,991,1684,2863,4867,8273,14064,23908, %T A034084 40643,69092,117457,199676,339449,577063,981007,1667712,2835110, %U A034084 4819686,8193466,13928892,23679116,40254498,68432646,116335497 %N A034084 Decimal part of n-th root of a(n) starts with digit 7. %e A034084 a(10)=202 -> 202^(1/10)=1.{7}0033751525... %Y A034084 Cf. A034064, A034074. %K A034084 nonn %O A034084 2,1 %A A034084 Patrick De Geest (pdg@worldofnumbers.com), Sep 1998. %I A018342 %S A018342 1,3,5,9,15,25,45,75,225 %N A018342 Divisors of 225. %K A018342 nonn,fini,full %O A018342 0,2 %A A018342 njas %I A029485 %S A029485 1,3,5,9,15,25,75,257,321,435,795,1285,2313,8523,39759,60855,91209, %T A029485 247875 %N A029485 n divides the (left) concatenation of all numbers <= n written in base 16 (most significant digit on left). %Y A029485 Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978. %K A029485 nonn,base %O A029485 1,2 %A A029485 Olivier Gerard (ogerard@ext.jussieu.fr) %E A029485 More terms from Olivier Gerard (ogerard@ext.jussieu.fr) %E A029485 More terms from Andrew Gacek (andrew@dgi.net), Feb 21 2000 and from Larry Reeves (larryr@acm.org), Aug 27 2001 %I A018436 %S A018436 1,3,5,9,15,27,45,81,135,405 %N A018436 Divisors of 405. %K A018436 nonn,fini,full %O A018436 0,2 %A A018436 njas %I A018298 %S A018298 1,3,5,9,15,27,45,135 %N A018298 Divisors of 135. %K A018298 nonn,fini,full %O A018298 0,2 %A A018298 njas %I A017913 %S A017913 1,1,3,5,9,15,27,46,81,140,243,420,729,1262,2187,3787,6561, %T A017913 11363,19683,34091,59049,102275,177147,306827,531441,920482, %U A017913 1594323,2761448,4782969,8284345,14348907,24853035,43046721 %N A017913 Powers of sqrt(3) rounded down. %K A017913 nonn %O A017913 0,3 %A A017913 njas %I A027154 %S A027154 1,1,3,5,9,15,31,49,101,163,331,533,1089,1751,3575,5753,11741, %T A027154 18891,38563,62045,126649,203775,415951,669249,1366101,2198003, %U A027154 4486651,7218853,14735409,23708711,48395175,77865993,158943181 %N A027154 a(n) = SUM{T(n-k,k)}, 0<=k<=[ n/2 ], T given by A027144. %K A027154 nonn %O A027154 0,3 %A A027154 Clark Kimberling, ck6@cedar.evansville.edu %I A052007 %S A052007 1,3,5,9,15,39,75,81,89,317,701,735,1311,1881,3201,3225 %N A052007 2^n + n is prime. %C A052007 Terms >= 701 are currently only strongpseudoprime. %e A052007 2^39 + 39 = 549755813927 is prime. %t A052007 Do[ If[ PrimeQ[ 2^n + n ], Print[ n ] ], {n, 0, 7000} ] %Y A052007 Cf. A048744. %K A052007 nonn,nice,hard %O A052007 1,2 %A A052007 G. L. Honaker, Jr. (curios@bvub.com) and Patrick De Geest (pdg@worldofnumbers.com), Nov 1999. %E A052007 Jud McCranie (jud.mccranie@mindspring.com) reports no other terms below 10000. %I A018260 %S A018260 1,3,5,9,15,45 %N A018260 Divisors of 45. %K A018260 nonn,fini,full %O A018260 0,2 %A A018260 njas %I A023656 %S A023656 0,1,3,5,9,16,27,45,74,120,195,317,514,833,1349,2184,3535,5721,9258, %T A023656 14980,24239,39221,63462,102685,166149,268836,434987,703825,1138814, %U A023656 1842641,2981457,4824100,7805559 %N A023656 Convolution of (F(2), F(3), F(4), ...) and A023534. %K A023656 nonn %O A023656 1,3 %A A023656 Clark Kimberling (ck6@cedar.evansville.edu) %I A018159 %S A018159 1,1,3,5,9,16,27,48,84,147,256,445,776,1351,2352,4096,7131, %T A018159 12416,21618,37640,65536,114104,198668,345901,602248,1048576, %U A018159 1825676,3178688,5534417,9635980,16777216,29210829,50859008 %N A018159 Powers of fifth root of 16 rounded down. %K A018159 nonn %O A018159 0,3 %A A018159 njas %I A054180 %S A054180 0,0,0,1,3,5,9,16,29,52,93,169,315,591,1119,2124,4052,7763,14922,28781, %T A054180 55629,107775,209241,406916,792579,1545812,3018687,5901609,11549484, %U A054180 22623452,44352970,87021370,170860941,335698520,659971004,1298228239 %N A054180 Number of positive integers <= 2^n of form 5 x^2 + 10 y^2. %K A054180 nonn %O A054180 0,5 %A A054180 dww %I A032679 %S A032679 3,5,9,17,23,27,29,35,39,41,45,69,77,81,87,101,111,129,135,153, %T A032679 167,171,173,183,191,195,197,207,215,231,233,237,245,249,279,293, %U A032679 299,311,321,333,339,341,345,357,359,371,377,395,401,405,419,435 %N A032679 Digit '7' concatenated with a(n) is a lucky number. %Y A032679 Cf. A000959. %K A032679 nonn %O A032679 0,1 %A A032679 Patrick De Geest (pdg@worldofnumbers.com), may 1998. %I A018162 %S A018162 1,1,3,5,9,17,29,52,93,163,289,509,897,1581,2787,4913,8658, %T A018162 15259,26891,47392,83521,147192,259403,457156,805665,1419857, %U A018162 2502271,4409853,7771662,13696315,24137569,42538611,74967511 %N A018162 Powers of fifth root of 17 rounded down. %K A018162 nonn %O A018162 0,3 %A A018162 njas %I A000213 M2454 N0975 %S A000213 1,1,1,3,5,9,17,31,57,105,193,355,653,1201,2209,4063,7473,13745,25281, %T A000213 46499,85525,157305,289329,532159,978793,1800281,3311233,6090307,11201821 %N A000213 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). %D A000213 B. G. Baumgart, Letter to the editor, Fib. Quart. 2 (1964), 260, 302. %D A000213 M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(#3) (1963), 71-74. %Y A000213 Cf. A000288, A000322, A000383, A046735, A060455. %K A000213 easy,nonn,nice %O A000213 0,4 %A A000213 njas %I A048578 %S A048578 3,5,9,17,33,65,129,257,513,1025,2049,4097,8193,16385,32769,65537,131073, %T A048578 262145,524289,1048577,2097153,4194305,8388609,16777217,33554433,67108865, %U A048578 134217729,268435457,536870913,1073741825,2147483649,4294967297,8589934593 %N A048578 Pisot sequence L(3,5). %H A048578 Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. %F A048578 a(n) = 2^(n+1)+1. a(n) = 3a(n-1) - 2a(n-2). %Y A048578 Subsequence of A000051. See A008776 for definitions of Pisot sequences. %K A048578 nonn %O A048578 0,1 %A A048578 dww %I A018095 %S A018095 1,3,5,9,17,35,71,143,289,587,1192,2420,4913,9977,20257, %T A018095 41133,83521,169593,344366,699250,1419857,2883081,5854221, %U A018095 11887248,24137569,49012377,99521747,202083205,410338673 %N A018095 Powers of fourth root of 17 rounded up. %K A018095 nonn %O A018095 0,2 %A A018095 njas %I A003217 M2455 %S A003217 3,5,9,17,35,79,209 %N A003217 Weights of threshold functions. %D A003217 S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 268. %K A003217 nonn %O A003217 2,1 %A A003217 njas %I A006723 M2456 %S A006723 1,1,1,1,1,1,1,3,5,9,17,41,137,769,1925,7203,34081,227321,1737001, %T A006723 14736001,63232441,702617001,8873580481,122337693603,1705473647525, %U A006723 22511386506929,251582370867257,9254211194697641,215321535159114017 %N A006723 Somos-7 sequence: a(n) = (a(n-1)a(n-6) + a(n-2)a(n-5) + a(n-3)a(n-4)) / a(n-7). %D A006723 R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115. %D A006723 David Gale, "The strange and surprising saga of the Somos sequence", Math. Intelligencer 13(1) (1991), pp. 40-42. %D A006723 J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261. %D A006723 R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619. %H A006723 S. Fomin and A. Zelevinsky, The Laurent phenomemon %H A006723 M. Somos, Somos 6 Sequence %H A006723 M. Somos, Brief history of the Somos sequence problem %H A006723 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A006723 Cf. A006720, A006721, A006722, A048736. %K A006723 nonn,easy,nice %O A006723 0,8 %A A006723 njas %E A006723 More terms from James A. Sellers (sellersj@math.psu.edu), Aug 22 2000 %I A062221 %S A062221 3,5,9,17,71,99,243,7595,9273 %N A062221 Smoothly undulating palindromic primes of the form (72*10^a(n)-27)/99. %C A062221 Prime versus probable prime status and proofs are given in the author's table. %H A062221 P. De Geest, SUPP Reference Table %e A062221 a(n)=17 -> (72*10^17-27)/99 = 72727272727272727. %Y A062221 Cf. A062209-A062232, A059758, A032758. %K A062221 nonn,base %O A062221 0,1 %A A062221 Patrick De Geest (pdg@worldofnumbers.com) and Hans Rosenthal (Hans.Rosenthal@t-online.de), Jun 15 2001. %I A028411 %S A028411 1,3,5,9,18,37,84,209,571 %N A028411 Number of types of Boolean functions of n variables under a certain group. %D A028411 I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167. %H A028411 Index entries for sequences related to Boolean functions %K A028411 nonn %O A028411 0,2 %A A028411 njas %E A028411 More terms from Vladeta Jovovic (vladeta@Eunet.yu) %I A018098 %S A018098 1,3,5,9,18,38,77,158,324,668,1375,2832,5832,12013,24744, %T A018098 50965,104976,216227,445376,917370,1889568,3892071,8016759, %U A018098 16512655,34012224,70057262,144301646,297227789,612220032 %N A018098 Powers of fourth root of 18 rounded up. %K A018098 nonn %O A018098 0,2 %A A018098 njas %I A032385 %S A032385 3,5,9,19,23,27,57,59,65,119,299,417,705,2255,4017,5193,5565,8055, %T A032385 17659,18039,19819,20507,21333,23289,103155,115913 %N A032385 71*2^n+1 is prime. %H A032385 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A032385 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A032385 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %H A032385 Y. Gallot, Proth.exe: Windows Program for Finding Large Primes %H A032385 R. Ballinger and W. Keller, More information %K A032385 nonn %O A032385 1,1 %A A032385 Jim Buddenhagen (jbuddenh@texas.net) %I A018002 %S A018002 1,3,5,9,19,39,81,169,351,729,1517,3155,6561,13648,28388, %T A018002 59049,122827,255491,531441,1105442,2299412,4782969,9948977, %U A018002 20694705,43046721,89540788,186252345,387420489,805867092 %N A018002 Powers of cube root of 9 rounded up. %K A018002 nonn %O A018002 0,2 %A A018002 njas %I A053662 %S A053662 3,5,9,21,23,33,39,51,63,65,81,89,95,99,113,131,173,183,191,209,215, %T A053662 221,239,245,251,261,281,285,299,303,309,315,341,345,363,369,371,393, %U A053662 411,419,431,443,473,495,509,525,543,545,561,575,593,645,659,683,711 %N A053662 2n+1 divides n!+1. %Y A053662 Cf. A005097, A053663. %K A053662 easy,nonn %O A053662 0,1 %A A053662 Chris K. Caldwell (caldwell@utm.edu), Feb 16 2000 %I A050355 %S A050355 1,1,3,5,9,21,27,81,37,81,111,297,201,243,513,1053,945,729,2187,1317, %T A050355 3645,365,2745,4077,2187,8829,7209,12393,2433,13257,16605,6561,34263, %U A050355 35397,41553,13473,59697,10155,64881,19683,44793,129033,18993,71307 %N A050355 Ordered factorizations with one level of parentheses indexed by prime signatures. A050354(A025487). %K A050355 nonn %O A050355 1,3 %A A050355 Christian G. Bower (bowerc@usa.net), Oct 1999. %I A006722 M2457 %S A006722 1,1,1,1,1,1,3,5,9,23,75,421,1103,5047,41783,281527,2534423,14161887, %T A006722 232663909,3988834875,45788778247,805144998681,14980361322965, %U A006722 620933643034787,16379818848380849,369622905371172929 %N A006722 Somos-6 sequence: a(n) = (a(n-1)a(n-5) + a(n-2)a(n-4) + a(n-3)^2)/a(n-6). %D A006722 R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115. %D A006722 David Gale, "The strange and surprising saga of the Somos sequence", Math. Intelligencer 13(1) (1991), pp. 40-42. %D A006722 J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261. %D A006722 C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 350. %D A006722 R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619. %H A006722 S. Fomin and A. Zelevinsky, The Laurent phenomemon %H A006722 M. Somos, Somos 6 Sequence %H A006722 M. Somos, Brief history of the Somos sequence problem %H A006722 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A006722 Cf. A006720, A006721, A006723, A048736. %K A006722 nonn,easy,nice %O A006722 0,7 %A A006722 njas %E A006722 More terms from James A. Sellers (sellersj@math.psu.edu), Aug 22 2000 %I A039774 %S A039774 3,5,9,25,31,57,116,144,154,288,372,414,624,792,10032 %N A039774 phi(a(n)) is equal to the product of (the sum of prime factors and the sum of exponents) of (a(n)-1). %C A039774 Next term if it exists is greater than 100000. %e A039774 phi(25)=20, 24=2^3*3^1, (2+3)*(3+1)=20. %Y A039774 Cf. A000010, A039697. %K A039774 nonn %O A039774 1,1 %A A039774 Olivier Gerard (ogerard@ext.jussieu.fr) %I A004044 %S A004044 1,3,5,9,27 %N A004044 Size of minimal covering code in {Z_3}^n with covering radius 1 (the next 2 terms are in the ranges 63-73, 153-186). %D A004044 G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 174. %D A004044 H. Hamalainen et al., Football pools - a game for mathematicians, Amer. Math. Monthly, 102 (1995), 579-588. %H A004044 Index entries for sequences related to covering codes %Y A004044 A column of A060439. %K A004044 nonn,hard,bref %O A004044 1,2 %A A004044 njas %I A013622 %S A013622 1,3,5,9,30,25,27,135,225,125,81,540,1350,1500,625,243,2025,6750, %T A013622 11250,9375,3125,729,7290,30375,67500,84375,56250,15625,2187,25515, %U A013622 127575,354375,590625,590625,328125,78125,6561,87480,510300,1701000 %N A013622 Triangle of coefficients in expansion of (3+5x)^n. %K A013622 tabl,nonn,easy %O A013622 0,2 %A A013622 njas %I A027715 %S A027715 3,5,9,33,383,555,3663,5115,30803,32223,34043,59295,567765,5912195, %T A027715 37877873,534141435,957747759,5356556535,5646996465,53205650235, %U A027715 56511511565,358023320853,9571923291759 %N A027715 Palindromes of form n^2+n+3. %H A027715 P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X) %Y A027715 Cf. A027714, A027688, A027752, A027753. %K A027715 nonn %O A027715 0,1 %A A027715 Patrick De Geest (pdg@worldofnumbers.com) %I A055289 %S A055289 1,1,1,0,3,5,9,44,50,545,572,9890,13263,251236,490208,8309897,25084529, %T A055289 340735734,1618199204,16683530000,123071639020,922641297931, %U A055289 10768715419165,53177799474462,1066100082041498 %V A055289 1,-1,1,0,-3,5,9,-44,-50,545,572,-9890,-13263,251236,490208,-8309897, %W A055289 -25084529,340735734,1618199204,-16683530000,-123071639020, %X A055289 922641297931,10768715419165,-53177799474462,-1066100082041498 %N A055289 Column 1 of triangle A055288. %H A055289 Index entries for sequences related to rooted trees %K A055289 sign,done %O A055289 1,5 %A A055289 Christian G. Bower (bowerc@usa.net), May 09 2000 %I A047389 %S A047389 3,5,10,12,17,19,24,26,31,33,38,40,45,47,52,54,59,61,66,68,73,75,80,82, %T A047389 87,89,94,96,101,103,108,110,115,117,122,124,129,131,136,138,143,145, %U A047389 150,152,157,159,164,166,171 %N A047389 Congruent to {3, 5} mod 7. %K A047389 nonn %O A047389 0,1 %A A047389 njas %I A007557 M2458 %S A007557 1,1,3,5,10,12,24,26,43,52,78,80,133,135,189,219,295,297, %T A007557 428,430,584,642,804,806,1100,1123,1395,1494,1856,1858,2428, %U A007557 2430,2977,3143,3739,3811,4790,4792,5654,5930,7072,7074,8656 %N A007557 Shifts left when inverse Moebius transform applied twice. %H A007557 M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Algebra and Its Applications, vol. 226-228, pp. 57-72, 1995 (Abstract, pdf, ps) %H A007557 N. J. A. Sloane, Transforms %K A007557 nonn,nice,eigen %O A007557 1,3 %A A007557 njas. %I A034746 %S A034746 1,3,5,10,12,26,26,49,59,95,118,221,270,457,682,1108,1650,2776,4242, %T A034746 7011,11082,17991,28736,46940,75149,121993,196659,318813,514336,833734, %U A034746 1346382,2180613,3525034,5706271,9227760,14936537,24157968,39096749 %N A034746 Dirichlet convolution of Fibonacci numbers with Primes (with 1). %K A034746 nonn %O A034746 1,2 %A A034746 Erich Friedman (erich.friedman@stetson.edu) %I A031878 %S A031878 0,1,3,5,10,13,21,25,36,41,55,61,78,85,105,113,136,145,171,181,210, %T A031878 221,253,265,300,313,351,365,406,421,465,481,528,545,595,613,666, %U A031878 685,741 %N A031878 Maximal number of edges in Hamiltonian PATH in complete graph on n nodes. %F A031878 C(n,2) if n odd, C(n,2)-n/2+1 if n even; G.f.: x^2*(1+2*x+x^3)/((1-x)*(1-x^2)). %e A031878 E.g. for n=4 [ 1:2 ][ 2:3 ][ 3:1 ][ 1:4 ][ 4:2 ], so a(4) = 5. %Y A031878 Cf. A031940. %K A031878 nonn %O A031878 1,3 %A A031878 Colin L. Mallows (colinm@research.avayalabs.com) %I A001767 M2459 N0976 %S A001767 0,0,0,0,1,3,5,10,13,26,25,50,49,73,81,133,109,196,169,241,241,375,289,476, %T A001767 421,568,529,806,577,1001,833,1081,1009,1393,1081,1768,1441,1849,1633 %N A001767 Genus of modular group GAMMA_n. %D A001767 R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 15. %D A001767 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 94. %H A001767 Index entries for sequences related to modular groups %K A001767 nonn,easy %O A001767 2,6 %A A001767 njas %I A048214 %S A048214 3,5,10,14,17,21,27,31,36,42,46,49,51,58,65,67,71,75,79,86,92,95,101, %T A048214 109,115,120,123,126,132,137,143,150,153,155,159,165,171,175,177,185, %U A048214 194,198,203,208,211,216,221,225,233,240,248,255 %N A048214 a(n)=T(n,2), array T given by A048212. %K A048214 nonn %O A048214 2,1 %A A048214 Clark Kimberling, ck6@cedar.evansville.edu %I A001841 M2460 N0977 %S A001841 3,5,10,14,21,26,36,43,55,64,78,88,105,117,136,150,171,186,210, %T A001841 227,253,272,300,320,351,373,406,430,465,490,528,555,595,624,666,696,741 %N A001841 Related to Zarankiewicz's problem. %D A001841 R. K. Guy, A problem of Zarankiewicz, in P. Erd\"{o}s and G. Katona, editors, Theory of Graphs (Proceedings of the Colloquium, Tihany, Hungary), Academic Press, NY, 1968, pp. 119-150, (p. 126, divided by 2). %K A001841 nonn %O A001841 3,1 %A A001841 njas %I A008610 %S A008610 1,1,3,5,10,14,22,30,43,55,73,91,116,140,172,204,245, %T A008610 285,335,385,446,506,578,650,735,819,917,1015,1128,1240, %U A008610 1368,1496,1641,1785,1947,2109,2290,2470,2670,2870,3091 %N A008610 Molien series of 4-dimensional representation of cyclic group of order 4 over GF(2) (not Cohen-Macaulay). %C A008610 a(n-4)=number of necklaces with 4 black beads and n-4 white beads. %C A008610 Also nonnegative integer 2x2 matrices with sum of elements equal to n, up to rotational symmetry. %D A008610 D. J. Benson, Poly. Invts. of Finite Grps, Cambr., 1993, p. 104. %H A008610 Index entries for sequences related to necklaces %e A008610 There are 10 inequivalent nonnegative integer 2x2 matrices with sum of elements equal to 4, up to rotational symmetry: %e A008610 [0 0] [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [0 2] [0 2] [1 1] %e A008610 [0 4] [1 3] [2 2] [3 1] [1 2] [2 1] [3 0] [1 1] [2 0] [1 1] . %p A008610 1/(1-x)/(1-x^2)^2/(1-x^4)*(1+2*x^3+x^4); %Y A008610 Cf. A000031, A047996, A005232, A008804. %K A008610 nonn,easy %O A008610 0,3 %A A008610 njas %E A008610 Comment and example from Vladeta Jovovic (vladeta@Eunet.yu), May 18 2000 %I A023602 %S A023602 1,3,5,10,15,20,29,37,47,58,70,85,100,116,134,153,173,195,219,244, %T A023602 268,296,324,354,387,419,454,489,526,564,603,645,689,734,779,825, %U A023602 874,923,973,1028,1082,1138,1196,1253,1313,1375,1436,1501,1567,1635 %N A023602 Convolution of A023532 and A000201. %K A023602 nonn %O A023602 1,2 %A A023602 Clark Kimberling (ck6@cedar.evansville.edu) %I A024452 %S A024452 1,3,5,10,15,23,31,42,55,69,86,105,125,148,173,200,230,262,296,331,369, %T A024452 409,452,498,547,597,649,702,757,819,883,950,1017,1090,1164,1240,1320,1402, %U A024452 1487,1574,1663,1757,1851,1948 %N A024452 a(n) = [ (2nd elementary symmetric function of P(n))/(1st elementary symmetric function of P(n)) ], where P(n) = {first n+1 primes}. %K A024452 nonn %O A024452 1,2 %A A024452 Clark Kimberling (ck6@cedar.evansville.edu) %I A045513 %S A045513 1,1,3,5,10,15,27,39,63,90,135,187,270,364,505,670,902, %T A045513 1173,1545,1976,2550,3218,4081,5083,6357,7825,9659,11772, %U A045513 14366,17342,20956,25080,30031,35667,42357,49945,58881 %N A045513 Expansion of 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)). %D A045513 Cohen, Arjeh M.; Griess, Robert L., Jr.; On finite simple subgroups of the complex Lie group of type $E\sb 8$. The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 367-405, Proc. Sympos. Pure Math., 47, Part 2, Amer. Math. Soc., Providence, RI, 1987. %K A045513 nonn %O A045513 0,3 %A A045513 njas %I A008337 %S A008337 1,1,3,5,10,15,27,40,63,91,134,188,265 %N A008337 Orbits on points that are at n steps from 0 in E_8 lattice. %H A008337 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (Abstract, pdf, ps). %H A008337 G. Nebe and N. J. A. Sloane, Home page for this lattice %K A008337 nonn,hard %O A008337 0,3 %A A008337 njas,jhc %I A054473 %S A054473 1,1,3,5,10,15,29,41,68,98,147,202,291,386,528,688,906,1151,1480,1841, %T A054473 2310,2833,3484,4207,5099,6076,7259,8562,10104,11796,13785,15948,18462, %U A054473 21201,24339,27747,31633,35827,40572,45695,51436,57618,64520,71918 %N A054473 Ways of numbering the faces of a cube so sum of the 6 numbers is n. %F A054473 G.f. : (3*x^6+x^5+x^4+1)/((1-x^4)*(1-x^3)^2*(1-x^2)^2*(1-x)). %Y A054473 Cf. A039959. %K A054473 easy,nonn %O A054473 0,3 %A A054473 Vladeta Jovovic (vladeta@Eunet.yu), May 20 2000 %I A006168 M2461 %S A006168 1,3,5,10,15,29,42,72,107,170,246,383,542,810,1145,1662,2311,3305,4537,6363 %N A006168 Factorization patterns of n. %D A006168 R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82. %K A006168 nonn %O A006168 1,2 %A A006168 njas %I A037246 %S A037246 1,0,0,1,1,1,3,5,10,16 %N A037246 Total number of fixed points in free homeomorphically irreducible trees with n nodes. %D A037246 F. Harary and E. M. Palmer, Probability that a point of a tree is fixed, Math. Proc. Camb. Phil. Soc. 85 (1979) 407-415. %H A037246 Index entries for sequences related to trees %F A037246 Reference gives a recurrence. %Y A037246 Cf. A005200-A005202. %K A037246 nonn,easy,more %O A037246 1,7 %A A037246 njas %I A032279 %S A032279 1,1,3,5,10,16,26,38,57,79,111,147,196,252,324,406,507,621,759, %T A032279 913,1096,1298,1534,1794,2093,2421,2793,3199,3656,4152,4706,5304, %U A032279 5967,6681,7467,8311,9234,10222,11298,12446,13691,15015,16445 %N A032279 Bracelets (turn over necklaces) of n beads of 2 colors, 5 of them black. %D A032279 S. J. Cyvin et al., Polygonal systems including the corannulene ... homologs ..., Z. Naturforsch., 52a (1997), 867-873. %H A032279 Index entries for sequences related to bracelets %H A032279 C. G. Bower, Transforms (2) %H A032279 F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. %F A032279 "DIK[ 5 ]" (necklace, indistinct, unlabeled, 5 parts) transform of 1,1,1,1... %F A032279 G.f.: (1-x+2*x^3-x^5+x^6)/((1-x)^2*(1-x^2)^2*(1-x^5)). %K A032279 nonn,easy,nice %O A032279 5,3 %A A032279 Christian G. Bower (bowerc@usa.net), njas %I A000990 M2462 N0978 %S A000990 1,1,3,5,10,16,29,45,75,115,181,271,413,605,895,1291,1866,2648, %T A000990 3760,5260,7352,10160,14008,19140,26085,35277,47575,63753,85175, %U A000990 113175,149938,197686,259891,340225,444135,577593,749131,968281 %N A000990 2-line partitions of n. %D A000990 L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.7). %D A000990 M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273. %D A000990 P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. %H A000990 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 141 %F A000990 G.f.: Product ( 1 - x^m )^(-2) (m=2..inf) / ( 1 - x ). %K A000990 nonn,easy %O A000990 0,3 %A A000990 njas %I A062773 %S A062773 3,5,10,16,31,40,62,73,100,147,163,220,264,284,330,410,488,520,610, %T A062773 676,706,812,887,1001,1164,1253,1295,1382,1424,1524,1878,1977,2142, %U A062773 2191,2490,2548,2730,2916,3044,3242,3437,3513,3869,3946,4090,4165,4628 %N A062773 Index of the smallest prime which follows n-th prime. %F A062773 a(n)=Pi[nextprime(p(n)^2)] %e A062773 100th prime, 541 immediately follows 529, square of 9th prime,a(9)=100. %Y A062773 A054270, A054271, A000879. %K A062773 nonn %O A062773 0,1 %A A062773 Labos E. (labos@ana1.sote.hu), Jul 18 2001 %I A005403 M2463 %S A005403 1,3,5,10,17,31,53,92,156,265 %N A005403 Protruded partitions of n. %D A005403 R. P. Stanley, A Fibonacci lattice, Fib. Quart., 13 (1975), 215-232. %K A005403 nonn %O A005403 1,2 %A A005403 njas %I A018072 %S A018072 1,1,3,5,10,17,31,56,100,177,316,562,1000,1778,3162,5623, %T A018072 10000,17782,31622,56234,100000,177827,316227,562341,1000000, %U A018072 1778279,3162277,5623413,10000000,17782794,31622776,56234132 %N A018072 Powers of fourth root of 10 rounded down. %K A018072 nonn %O A018072 0,3 %A A018072 njas %I A054166 %S A054166 0,0,1,1,3,5,10,17,32,56,106,195,365,691,1308,2495,4773,9174,17671, %T A054166 34144,66131,128330,249482,485761,947206,1849299,3614605,7072464, %U A054166 13851148,27150649,53262111,104562496,205412762,403786919,794202162 %N A054166 Number of positive integers <= 2^n of form 3 x^2 + 9 y^2. %K A054166 nonn %O A054166 0,5 %A A054166 dww %I A054157 %S A054157 0,1,1,3,5,10,17,33,58,109,202,379,720,1369,2618,5031,9701,18766,36387, %T A054157 70688,137572,268208,523581,1023425,2002829,3923400,7692930,15096706, %U A054157 29649082,58270386,114596178,225503077,443996993,874648723,1723849033 %N A054157 Number of positive integers <= 2^n of form 2 x^2 + 7 y^2. %K A054157 nonn %O A054157 0,4 %A A054157 dww %I A026621 %S A026621 1,1,3,5,10,17,34,60,120,217,434,798,1596,2970,5940,11154,22308, %T A026621 42185,84370,160446,320892,613054,1226108,2351440,4702880,9048522, %U A026621 18097044,34916300,69832600,135059220,270118440,523521630,1047043260 %N A026621 T(n,[ n/2 ]), T given by A026615. %K A026621 nonn %O A026621 0,3 %A A026621 Clark Kimberling, ck6@cedar.evansville.edu %I A026687 %S A026687 1,1,3,5,10,17,44,87,174,324,822,1707,3414,6595,16604,35300,70600, %T A026687 139114,348828,751901,1503802,3000177,7504156,16322561,32645122, %U A026687 65684626,164014374,359049028,718098056,1453825024,3625748104 %N A026687 T(n,[ n/2 ]), T given by A026681. %K A026687 nonn %O A026687 0,3 %A A026687 Clark Kimberling, ck6@cedar.evansville.edu %I A009854 %S A009854 1,3,5,10,18,29,45,56,65,86,110,130,151,177,210,246,292,344,384,430, %T A009854 491,541,581,630,690,750,805,882,985,1069,1138,1225,1313,1378,1441, %U A009854 1542,1654,1725,1794,1895,1992,2092,2226,2377,2503,2596,2708,2842,2951 %N A009854 Coordination sequence T5 for Zeolite Code -CLO. %D A009854 W. M. Meier, D. H. Olson and Ch. Baerlocher, Atlas of Zeolite Structure Types, 4th Ed., Elsevier, 1996 %H A009854 R. W. Grosse-Kunstleve, Coordination Sequences and Encyclopedia of Integer Sequences %H A009854 R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites, Acta Cryst., A52 (1996), pp. 879-889. %H A009854 International Zeolite Association, Database of Zeolite Structures %K A009854 nonn %O A009854 0,2 %A A009854 rwgk@cci.lbl.gov (R.W. Grosse-Kunstleve) %I A018165 %S A018165 1,1,3,5,10,18,32,57,101,181,324,577,1029,1835,3271,5832, %T A018165 10396,18532,33035,58889,104976,187130,333579,594639,1060005, %U A018165 1889568,3368348,6004426,10703505,19080094,34012224,60630274 %N A018165 Powers of fifth root of 18 rounded down. %K A018165 nonn %O A018165 0,3 %A A018165 njas %I A054179 %S A054179 0,0,0,1,3,5,10,18,32,60,110,207,391,739,1411,2695,5158,9941,19187, %T A054179 37151,72038,140000,272470,531133,1036727,2025812,3962913,7759632, %U A054179 15207510,29828260,58549403,115004444,226040861,444547031,874762315 %N A054179 Number of positive integers <= 2^n of form 5 x^2 + 9 y^2. %K A054179 nonn %O A054179 0,5 %A A054179 dww %I A010049 %S A010049 0,1,1,3,5,10,18,33,59,105,185,324,564,977,1685,2895,4957, %T A010049 8462,14406,24465,41455,70101,118321,199368,335400,563425, %U A010049 945193,1583643,2650229,4430290,7398330,12342849,20573219 %N A010049 Second-order Fibonacci numbers. %D A010049 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 83. %D A010049 L. Turban, Lattice animals on a staircase and Fibonacci numbers, J.Phys. A 33 (2000) 2587-2595. %F A010049 a(n)=((2*n+3)*F(n)-n*F(n-1))/5, F(n)=A000045(n) (Fibonacci numbers) (Turban ref. eq.(2.12)). %F A010049 G.f.: x*(1-x)/(1-x-x^2)^2 (Turban ref. eq.(2.10)). %p A010049 with(combinat): A010049 := proc(n) options remember; if n <= 1 then n else A010049(n-1)+A010049(n-2)+fibonacci(n-2); fi; end; %K A010049 nonn %O A010049 0,4 %A A010049 njas %E A010049 Additional comments from Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), May 03 2000 %I A001445 %S A001445 3,5,10,18,36,68,136,264,528,1040,2080,4128,8256,16448, %T A001445 32896,65664,131328,262400,524800,1049088,2098176,4195328, %U A001445 8390656,16779264,33558528,67112960,134225920,268443648 %N A001445 (2^n + 2^[ n/2 ] )/2. %p A001445 f:=n->(2^n+2^floor(n/2))/2; %K A001445 nonn %O A001445 2,1 %A A001445 njas %I A018168 %S A018168 1,1,3,5,10,19,34,61,111,200,361,650,1172,2112,3806,6859, %T A018168 12359,22272,40134,72320,130321,234836,423170,762546,1374096, %U A018168 2476099,4461888,8040248,14488391,26107836,47045881,84775883 %N A018168 Powers of fifth root of 19 rounded down. %K A018168 nonn %O A018168 0,3 %A A018168 njas %I A014610 %S A014610 3,5,10,19,37,71,137,264,509,981,1891,3645,7026,13543,26105, %T A014610 50319,96993,186960,360377,694649,1338979,2580965,4974970, %U A014610 9589563,18484477,35629975,68678985,132383000,255176437 %N A014610 Tetranacci numbers arising in connection with current algebras sp(2)_n. %H A014610 Reference %K A014610 nonn %O A014610 0,1 %A A014610 Nouredine Chair (chair@sissa.it) %I A003055 M2464 %S A003055 1,1,3,5,10,19,39 %N A003055 Connected graphs that can be drawn in the plane using unit-length edges. %C A003055 K_4 can't be so drawn even though it is planar. %D A003055 M. Gardner, The Unexpected Hanging and Other Mathematical Diversions. Simon and Schuster, NY, 1969, p. 80. %H A003055 E. W. Weisstein, Link to a section of The World of Mathematics. %K A003055 nonn,more,nice %O A003055 1,3 %A A003055 njas %E A003055 Brendan McKay (bdm@cs.anu.edu.au) points out that this is wrong. On page 80 there are two graphs missing from figure 32 (a triangle with a tail of length two, and one of the trees). So the sequence starts 1,1,3,5,12. The next number isn't 19 as Figure 37 on page 87 misses 4 trees (at least). This sequence should be recomputed! %I A018101 %S A018101 1,3,5,10,19,40,83,173,361,754,1574,3286,6859,14321,29898, %T A018101 62421,130321,272084,568057,1185987,2476099,5169594,10793066, %U A018101 22533737,47045881,98222279,205068241,428140988,893871739 %N A018101 Powers of fourth root of 19 rounded up. %K A018101 nonn %O A018101 0,2 %A A018101 njas %I A018104 %S A018104 1,3,5,10,20,43,90,190,400,846,1789,3783,8000,16918,35778, %T A018104 75660,160000,338359,715542,1513187,3200000,6767177,14310836, %U A018104 30263732,64000000,135343522,286216702,605274630,1280000000 %N A018104 Powers of fourth root of 20 rounded up. %K A018104 nonn %O A018104 0,2 %A A018104 njas %I A037183 %S A037183 3,5,10,21,36,60,80,120,180,264,252,360,300,960,900,720,1080,1440,1800,1680, %T A037183 2160,2880,5616,3780,2520,3600,6120,6720,6300,5040,11340,7560,14112,10800,9240, %U A037183 10080,13860,12600,31200,15120,22680,20160,18480,39312,33264,39600,25200,30240 %N A037183 Smallest number which is palindromic (with at least 2 digits) in n bases. %K A037183 nonn,base,nice %O A037183 1,1 %A A037183 Erich Friedman (erich.friedman@stetson.edu) %E A037183 More terms from dww %I A024424 %S A024424 0,0,1,1,3,5,10,21,42,63,190,297,484,1144,2431,4004,10868,12155,39117,54587, %T A024424 121771,205751,529074,1337220,1961256,1634380,8469056,17829888 %N A024424 a(n) = greatest residue of S(n,m) mod C(n-1,m-1), for m = 1,2,...,n; S(n,m) are Stirling numbers of second kind. %K A024424 nonn %O A024424 1,5 %A A024424 Clark Kimberling (ck6@cedar.evansville.edu) %I A018107 %S A018107 1,3,5,10,21,45,97,207,441,945,2021,4327,9261,19825,42440, %T A018107 90850,194481,416325,891224,1907839,4084101,8742816,18715702, %U A018107 40064613,85766121,183599119,393029742,841356859,1801088541 %N A018107 Powers of fourth root of 21 rounded up. %K A018107 nonn %O A018107 0,2 %A A018107 njas %I A053709 %S A053709 3,5,10,21,171,190 %N A053709 Prime balanced factorials: n! is the mean of its 2 closest primes. %e A053709 For the 2nd term 5! is in the middle between closest neighbors 113 and 127. %p A053709 for n from 3 to 200 do j:=n!-prevprime(n!): if not isprime(n!+j) then next fi: i:=1: while not isprime(n!+i) and (i<=j) do i:=i+2 od: if i=j then print(n):fi:od: %Y A053709 A033393, A033392, A006990, A037151, A006562, A053710. %K A053709 more,nonn %O A053709 1,1 %A A053709 Labos E. (labos@ana1.sote.hu), Feb 10 2000 %E A053709 171 and 190 from Jud McCranie (jud.mccranie@mindspring.com), Jul 04 2000 %I A018005 %S A018005 1,3,5,10,22,47,100,216,465,1000,2155,4642,10000,21545, %T A018005 46416,100000,215444,464159,1000000,2154435,4641589,10000000, %U A018005 21544347,46415889,100000000,215443470,464158884,1000000000 %N A018005 Powers of cube root of 10 rounded up. %K A018005 nonn %O A018005 0,2 %A A018005 njas %I A061438 %S A061438 0,3,5,10,22,47,100,216,465,1000,2155,4642,10000,21545,46416,100000, %T A061438 215444,464159,1000000,2154435,4641589,10000000,21544347,46415889, %U A061438 100000000,215443470,464158884,1000000000,2154434691,4641588834 %N A061438 Smallest number whose cube has n digits. %e A061438 a(5) = 22, 22^3 = 10648 has 5 digits, while 21^3 = 9261 has 4 digits. %Y A061438 Cf. A061434, A061439. %K A061438 nonn,base,easy %O A061438 1,2 %A A061438 Amarnath Murthy (amarnath_murthy@yahoo.com), May 03 2001 %E A061438 More terms from Larry Reeves (larryr@acm.org), May 16 2001 %I A002039 M2465 N0979 %S A002039 1,3,5,10,25,64,160,390,940,2270,5515,13440,32735,79610,193480, %T A002039 470306,1143585,2781070,6762990,16445100,39987325,97232450,236432060, %U A002039 574915770,1397981470,3399360474,8265943685,20099618590,48874630750 %N A002039 Related to partitions (g.f. is inverse to A000203). %D A002039 J. M. Gandhi, On numbers related to partitions of a number, Amer. Math. Monthly, 76 (1969), 1033-1036. %F A002039 G.f.: Sum (-1)^n*a(n)*x^n = -F(x)/F'(x), F(x) = Product (1-x^k), k=1..inf. %Y A002039 Cf. A002040, A000203. %K A002039 nonn,nice,easy %O A002039 0,2 %A A002039 njas, sp. %I A007695 M2466 %S A007695 3,5,10,26,96,553,5461,100709,3718354,289725509,49513793526, %T A007695 19089032278261,16951604697397302,35231087224279091310, %U A007695 173550485517380958360611,2047581288200721764035942914 %N A007695 Cardinalities of Sperner families on 1,...,n. %D A007695 S. Johnson, Upper bounds for constant weight error correcting codes, Discrete Math., 3 (1972), 109-124. %D A007695 D. E. Knuth, Art of Computer Programming, Vol. 4, Section 7.3. %t A007695 c[ 0,0 ]=1; c[ 0,1 ]=1; kap[ 0,0 ]=0; f[ n_ ]:=Block[ {s=2,r,d,k,j}, For[ r=1,r<=n,r++, d=s; k=r; j=0; s=0; %t A007695 For[ x=0, x<=Binomial[ n,r ], x++, If[ x>=Binomial[ k,r ],k++,0 ]; kap[ r,x ]=If[ x==0,0,Binomial[ k-1,r-1 ]+kap[ r-1,x-Binomial[ k-1,r ] ] ]; %t A007695 While[ j, Cycle indices of linear, affine and projective groups, Linear Algebra and Its Applications, 263, 133-156, 1997. %H A000214 H. Fripertinger, Implementation of cycle index of linear group %H A000214 Vladeta Jovovic, Cycle indices %H A000214 Index entries for sequences related to Boolean functions %Y A000214 Cf. A000585. %K A000214 nonn,nice,easy,huge %O A000214 1,1 %A A000214 njas %E A000214 More terms from Vladeta Jovovic (vladeta@Eunet.yu) %I A060955 %S A060955 1,1,1,3,5,10,35,70,216,567,2310,5775,21450,69498 %N A060955 Highest degree of an irreducible representation of the alternating group A_n of degree n. %Y A060955 A003040, A003861-A003868. %K A060955 nonn %O A060955 1,4 %A A060955 Ahmed Fares (ahmedfares@my-deja.com), May 08 2001 %I A024329 %S A024329 0,0,1,3,5,10,116,126,6054,68760,255336,3099360,53045136,55156608, %T A024329 10024873104,175247246160,693221858736,29064929199744,605379922107648, %U A024329 445102785680256,264990555297131904,5736069117714297600 %V A024329 0,0,1,-3,5,10,-116,126,6054,-68760,255336,3099360,-53045136,55156608, %W A024329 10024873104,-175247246160,693221858736,29064929199744,-605379922107648, %X A024329 -445102785680256,264990555297131904,-5736069117714297600 %N A024329 Expansion of ln(1+ln(1+x)^2)/2. %t A024329 Log[ 1+Log[ 1+x ]^2 ]/2 %Y A024329 A009325. %K A024329 sign,done %O A024329 0,4 %A A024329 rhh@research.bell-labs.com %E A024329 Extended with signs 03/97. %I A064523 %S A064523 1,3,5,10,2640,8304,11699,15330,16810,16910,22463,25906,26652,35950, %T A064523 72429,75470,270353,296073,371964 %N A064523 pi(n^2) is a square. %e A064523 n = 14: a(14) = 35950, Pi(35950^2) = Pi(1292402500) = 64866916 = 8054^2 %t A064523 Do[ If[ IntegerQ[ Sqrt[ PrimePi[n^2]]], Print[n]], {n, 1, 400000} ] %o A064523 (PARI) pi(x, c=0) = forprime(p=2,x,c++);c for(n=1,10^6,if(issquare(pi(n^2)),print1(n," "))) %Y A064523 Cf. A000720. %K A064523 nonn %O A064523 1,2 %A A064523 Jason Earls (jcearls@kskc.net), Oct 07 2001 %E A064523 More terms from Robert G. Wilson v (rgwv@kspaint.com) and Labos E. (labos@ana1.sote.hu), Oct 08 2001 %E A064523 Further terms from Robert G. Wilson v (rgwv@kspaint.com), Oct 16 2001 %I A046228 %S A046228 3,5,11,7,9,9,20,27,11,67,19,13,147,105,51,15,126,78,33,17,273,204,111, %T A046228 83,19,477,315,305,51,21,1023,792,935,407,123,23,1815,2519,671,265,73, %U A046228 25,3861,6149,3861,936,338,171,27,13871,5005,5733,1274,847,99,29,29315 %N A046228 First numerator and then denominator of elements to right of central elements of 1/2-Pascal triangle (by row), excluding 1's and 2's. %e A046228 1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ... %Y A046228 Cf. A046213. %K A046228 tabf,nonn %O A046228 1,1 %A A046228 Mohammad K. Azarian, ma3@cedar.evansville.edu %E A046228 More terms from James A. Sellers (sellersj@math.psu.edu), Dec 13 1999 %I A065019 %S A065019 1,3,5,11,11,13,15,17,19,21,25,27,29,31,35,35,39,41,45,49,49,51,53,55, %T A065019 57,61,63,65,67,69,73,75,77,81,83,83,87,91,95,95,99,99,103,103,105,107, %U A065019 113,113,115,117,121,123,125,129,131,133,135,137,139,141,143,147,149 %N A065019 Let phi be the golden number {1+sqrt(5)}/2 (A001622), let phi(n) be the number phi written in base 10 but truncated to n decimal digits. Sequence gives number of 1's at the beginning of the continued fraction expansion of phi(n). %C A065019 a(n) has the curious property of always being odd but is otherwise quite random. Nevertheless c = lim(n-->infinity) a(n)/n exists, about 2.3926 +/- 0.0004. %F A065019 The value of lim n -> infinity a(n)/n is ln(10)/2/ln(phi)=2,3924... %e A065019 phi(6)=1.618033. The continued fraction expansion of phi(6) = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 129}. Hence a(6) = 15. %t A065019 gr = RealDigits[ N[ GoldenRatio, 250]] [[1]]; f[n_] := Block[ {k = 1}, While[ ContinuedFraction[ FromDigits[ {Take[ gr, n + 1 ], 1} ]] [[k]] == 1, k++ ]; k - 1]; Table[ f[n], {n, 0, 70} ] %Y A065019 Cf. A001622. %K A065019 nonn %O A065019 0,2 %A A065019 Benoit Cloitre (abcloitre@wanadoo.fr) and Boris Gourevitch (boris@314.net), Nov 02 2001 %E A065019 Additional comments from Robert G. Wilson v (rgwv@kspaint.com), Nov 02 2001 %I A006538 M2471 %S A006538 1,3,5,11,11,19,35,47,53,95,103,179,251,299,503,743,1019,1319,1439,2939,3359, %T A006538 3959,5387,5387,5879,5879,17747,17747,23399,23399,23399,23399,23399,23399,93596 %N A006538 Worst cases for Pierce expansions. %D A006538 Erdos, P.; Shallit, J. O.; New bounds on the length of finite Pierce and Engel series. Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53. %D A006538 M. E. Mays, Iterating the division algorithm, Fib. Quart., 25 (1987), 204-213. %H A006538 Index entries for sequences related to Engel expansions %K A006538 nonn %O A006538 1,2 %A A006538 jos,njas %E A006538 Description corrected 5/95. %I A065396 %S A065396 3,5,11,13,17,19,23,29,31,37,41,43,47,53,59,67,71,73,79,83,89,97,101, %T A065396 103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191, %U A065396 193,197,199,223,227,229,233,239,241,251,257,263,269,271,277,281,283 %N A065396 Primes of the form p + k*(k+1) / 2, p prime and k > 0. %C A065396 a(2) = 11 = 5 + 3*(3+1) / 2. %Y A065396 A000040, A000217, A065376, A065397. %K A065396 nonn %O A065396 0,1 %A A065396 Reinhard Zumkeller (reinhard.zumkeller@lhsystems.com), Nov 05 2001 %I A020620 %S A020620 3,5,11,13,17,19,23,29,31,37,41,47,53,61,67,71,83,97,101,109,163,181,191, %T A020620 239,263,293,307,383,431,691,821,2111,2351 %N A020620 Smallest nonempty set S containing prime divisors of 8k+7 for each k in S. %K A020620 nonn,fini,full %O A020620 1,1 %A A020620 dww %I A059636 %S A059636 3,5,11,13,17,19,23,29,37,41,43,53,59,61,67,83,89,97,101,107,109,131, %T A059636 137,139,149,157,163,173,179,181,193,197,199,211,227,229,241,251,269, %U A059636 277,283,293,307,313,317,331,347,349,353,373,379,389,397,401,409,419 %N A059636 Primes p such that x^44 = 2 has no solution mod p. %C A059636 Complement of A049576 relative to A000040. %Y A059636 A000040, A049576. %K A059636 nonn %O A059636 0,1 %A A059636 Klaus Brockhaus (klaus-brockhaus@t-online.de), Feb 02 2001 %I A059352 %S A059352 3,5,11,13,17,19,29,31,37,41,43,53,59,61,67,71,83,97,101,107,109,113, %T A059352 131,137,139,149,157,163,173,179,181,191,193,197,211,227,229,241,251, %U A059352 269,271,277,281,283,293,307,311,313,317,331,347,349,353,373,379,389 %N A059352 Primes p such that x^40 = 2 has no solution mod p. %C A059352 Complement of A049572 relative to A000040. %Y A059352 A000040, A049572. %K A059352 nonn %O A059352 0,1 %A A059352 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 27 2001 %I A059309 %S A059309 3,5,11,13,17,19,29,31,37,41,43,53,59,61,67,71,83,97,101,107,109,131, %T A059309 137,139,149,157,163,173,179,181,191,193,197,211,227,229,241,251,269, %U A059309 271,277,281,283,293,307,311,313,317,331,347,349,373,379,389,397,401 %N A059309 Primes p such that x^20 = 2 has no solution mod p. %C A059309 Complement of A049552 relative to A000040. %Y A059309 A000040, A049552. %K A059309 easy,nonn %O A059309 0,1 %A A059309 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 25 2001 %I A049233 %S A049233 3,5,11,13,17,19,29,31,37,41,53,59,67,71,83,89,101,103,107,109,113,127, %T A049233 131,137,139,149,157,163,179,181,191,193,197,199,211,227,229,233,239, %U A049233 251,257,263,269,271,281,283,293,307,311,317,337,347,353,379,383,389 %N A049233 p_k + 2 is square-free, where p_k is the k-th prime. %F A049233 Abs[ MoebiusMu[ Prime[ k ]+2 ]=1 %K A049233 nonn %O A049233 1,1 %A A049233 Labos E. (labos@ana1.sote.hu) %I A045402 %S A045402 3,5,11,13,17,19,29,31,41,43,47,53,59,61,67,71,73,83,89,97,101,103,109, %T A045402 113,127,131,137,139,151,157,167,173,179,181,193,197,199,211,223,227, %U A045402 229,239,241,251,257,263,269 %N A045402 Primes congruent to {1, 3, 4, 5, 6} mod 7. %K A045402 nonn %O A045402 0,1 %A A045402 njas %I A059634 %S A059634 3,5,11,13,17,19,29,37,41,43,53,59,61,67,71,83,97,101,107,109,113,127, %T A059634 131,137,139,149,157,163,173,179,181,193,197,211,227,229,239,241,251, %U A059634 269,277,281,283,293,307,313,317,331,337,347,349,353,373,379,389,397 %N A059634 Primes p such that x^56 = 2 has no solution mod p. %C A059634 Complement of A049588 relative to A000040. Coincides for the first 51 terms with sequence A059315 of primes p such that x^28 = 2 has no solution mod p (first divergence is at 353, cf. A059635). %Y A059634 A000040, A049588, A059315, A059635. %K A059634 nonn %O A059634 0,1 %A A059634 Klaus Brockhaus (klaus-brockhaus@t-online.de), Feb 02 2001 %I A059315 %S A059315 3,5,11,13,17,19,29,37,41,43,53,59,61,67,71,83,97,101,107,109,113,127, %T A059315 131,137,139,149,157,163,173,179,181,193,197,211,227,229,239,241,251, %U A059315 269,277,281,283,293,307,313,317,331,337,347,349,373,379,389,397,401 %N A059315 Primes p such that x^28 = 2 has no solution mod p. %C A059315 Coincides for the first 51 terms with sequence A059634 of primes p such that x^56 = 2 has no solution mod p (first divergence is at 353, cf. A059635 ). %C A059315 Complement of A049560 relative to A000040. %Y A059315 A000040, A049560, A059634, A059635. %K A059315 nonn %O A059315 0,1 %A A059315 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 25 2001 %I A045403 %S A045403 3,5,11,13,17,19,29,37,41,43,53,59,61,67,73,83,89,97,101,107,109,113, %T A045403 131,137,139,149,157,163,173,179,181,193,197,211,227,229,233,241,251, %U A045403 257,269,277,281,283,293,307,313 %N A045403 Primes congruent to {1, 3, 5} mod 8. %K A045403 nonn %O A045403 0,1 %A A045403 njas %I A059641 %S A059641 3,5,11,13,17,19,29,37,41,43,53,59,61,67,79,83,97,101,107,109,131,137, %T A059641 139,149,157,163,173,179,181,193,197,211,227,229,241,251,269,277,283, %U A059641 293,307,313,317,331,347,349,373,379,389,397,401,409,419,421,433,443 %N A059641 Primes p such that x^52 = 2 has no solution mod p. %C A059641 Complement of A049584 relative to A000040 %Y A059641 A000040, A049584. %K A059641 nonn %O A059641 0,1 %A A059641 Klaus Brockhaus (klaus-brockhaus@t-online.de), Feb 02 2001 %I A014662 %S A014662 3,5,11,13,17,19,29,37,41,43,53,59,61,67,83,97,101,107, %T A014662 109,113,131,137,139,149,157,163,173,179,181,193,197,211, %U A014662 227,229,241,251,257,269,277,281,283,293,307,313,317,331 %N A014662 Primes p such that order of 2 mod p is even. %D A014662 P. Moree, Appendix to V. Pless et al., Cyclic Self-Dual Z_4 Codes, Finite Fields Applic., vol 3 pp 48-69, 1997. %K A014662 nonn %O A014662 0,1 %A A014662 njas %I A059349 %S A059349 3,5,11,13,17,19,29,37,41,43,53,59,61,67,83,97,101,107,109,113,131,137, %T A059349 139,149,157,163,173,179,181,193,197,211,227,229,241,251,257,269,277, %U A059349 281,283,293,307,313,317,331,347,349,353,373,379,389,397,401,409,419 %N A059349 Primes p such that x^32 = 2 has no solution mod p. %C A059349 Complement of A049564 relative to A000040. %Y A059349 A000040, A049564. %K A059349 nonn %O A059349 0,1 %A A059349 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 27 2001 %I A045316 %S A045316 3,5,11,13,17,19,29,37,41,43,53,59,61,67,83,97,101,107,109,113,131,137, %T A045316 139,149,157,163,173,179,181,193,197,211,227,229,241,251,269,277,281, %U A045316 283,293,307,313,317,331,347 %N A045316 Primes p such that x^8 = 2 has no solution mod p. %C A045316 Complement of A045315 relative to A000040. Coincides for the first 140 terms with the sequence of primes p such that x^16 = 2 has no solution mod p (first divergence is at 1217, cf. A059287) - Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 26 2001. %Y A045316 Cf. A000040, A045315, A059287. %K A045316 nonn %O A045316 0,1 %A A045316 njas %I A040100 %S A040100 3,5,11,13,17,19,29,37,41,43,53,59,61,67,83,97,101,107,109,131,137,139, %T A040100 149,157,163,173,179,181,193,197,211,227,229,241,251,269,277,283,293, %U A040100 307,313,317,331,347,349,373 %N A040100 Primes p such that x^4 = 2 has no solution mod p. %K A040100 nonn %O A040100 0,1 %A A040100 njas %I A045404 %S A045404 3,5,11,13,17,19,31,41,47,53,59,61,67,73,83,89,97,101,103,109,131,137, %T A045404 139,151,157,167,173,179,181,193,199,223,227,229,241,251,257,263,269, %U A045404 271,277,283,293,307,311,313,347 %N A045404 Primes congruent to {3, 4, 5, 6} mod 7. %K A045404 nonn %O A045404 0,1 %A A045404 njas %I A020612 %S A020612 3,5,11,13,17,23,29,43,103,127,211 %N A020612 Smallest nonempty set S containing prime divisors of 7k+8 for each k in S. %K A020612 nonn,fini,full %O A020612 1,1 %A A020612 dww %I A062391 %S A062391 3,5,11,13,17,23,31,43,53,61,67,71,73,79,89,101,103,107,127,139,167, %T A062391 173,181,193,197,211,223,227,233,241,269,277,281,349,353,359,379,433, %U A062391 467,499,521,523,557,577,587,613,631,743,757,769,821,827,829,883,947 %N A062391 a(1) = 3, a(2) = 5; a(n+1) = smallest prime number such that the sum of any three consecutive terms is a prime. %e A062391 After 43, the next term is 53, since 31+43+47=121 is not prime and 31+43+53=127 is prime. %K A062391 nonn,easy %O A062391 0,1 %A A062391 Amarnath Murthy (amarnath_murthy@yahoo.com), Jun 27 2001 %E A062391 Corrected and extended by Larry Reeves (larryr@acm.org), Jul 02 2001 %I A038951 %S A038951 3,5,11,13,17,23,31,53,73,83,89,107,113,127,137,139,149, %T A038951 151,163,191,193,211,223,227,251,263,271,277,281,293,307, %U A038951 331,349,359,383,389,397,401,409,419,431,439,463,467,479 %N A038951 69 is a square mod p. %K A038951 nonn %O A038951 0,1 %A A038951 njas %I A020578 %S A020578 3,5,11,13,17,29,37,41,89 %N A020578 Smallest nonempty set S containing prime divisors of 2k+7 for each k in S. %K A020578 nonn,fini,full %O A020578 1,1 %A A020578 dww %I A047621 %S A047621 3,5,11,13,19,21,27,29,35,37,43,45,51,53,59,61,67,69,75,77,83,85,91,93, %T A047621 99,101,107,109,115,117,123,125,131,133,139,141,147,149,155,157,163, %U A047621 165,171,173,179,181,187,189 %N A047621 Congruent to {3, 5} mod 8. %K A047621 nonn %O A047621 0,1 %A A047621 njas %I A045405 %S A045405 3,5,11,13,19,23,29,31,41,43,53,59,61,71,73,79,83,89,101,103,109,113, %T A045405 131,139,149,151,163,173,179,181,191,193,199,211,223,229,233,239,241, %U A045405 251,263,269,271,281,283,293,311 %N A045405 Primes congruent to {0, 1, 3, 4} mod 5. %K A045405 nonn %O A045405 0,1 %A A045405 njas %I A059311 %S A059311 3,5,11,13,19,23,29,37,43,53,59,61,67,83,89,101,107,109,131,139,149, %T A059311 157,163,173,179,181,197,199,211,227,229,251,269,277,283,293,307,317, %U A059311 331,347,349,353,373,379,389,397,419,421,443,461,463,467,491,499,509 %N A059311 Primes p such that x^22 = 2 has no solution mod p. %C A059311 Complement of A049554 relative to A000040. %Y A059311 A000040, A049554. %K A059311 easy,nonn %O A059311 0,1 %A A059311 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 25 2001 %I A023209 %S A023209 3,5,11,13,19,23,31,41,53,59,73,79,89,101,103,109,131,139,151,173,179, %T A023209 181,191,199,223,229,241,251,269,283,293,311,331,349,353,373,383,389,409, %U A023209 431,433,439,509,521,541,563,593,599,619,643,661,683,709,719,733,739,761 %N A023209 n and 3n + 4 both prime. %K A023209 nonn %O A023209 1,1 %A A023209 dww %I A020609 %S A020609 3,5,11,13,19,23,107,137,251,457,587,3203 %N A020609 Smallest nonempty set S containing prime divisors of 7k+4 for each k in S. %K A020609 nonn,fini,full %O A020609 1,1 %A A020609 dww %I A059639 %S A059639 3,5,11,13,19,29,31,37,41,43,53,59,61,67,71,83,101,107,109,131,139,149, %T A059639 151,157,163,173,179,181,191,197,211,227,229,251,269,271,277,281,283, %U A059639 293,307,311,317,331,347,349,373,379,389,397,401,419,421,443,461,467 %N A059639 Primes p such that x^50 = 2 has no solution mod p. %C A059639 Complement of A049582 relative to A000040 %Y A059639 A000040, A049582. %K A059639 nonn %O A059639 0,1 %A A059639 Klaus Brockhaus (klaus-brockhaus@t-online.de), Feb 02 2001 %I A059263 %S A059263 3,5,11,13,19,29,31,37,41,43,53,59,61,67,71,83,101,107,109,131,139,149, %T A059263 157,163,173,179,181,191,197,211,227,229,251,269,271,277,281,283,293, %U A059263 307,311,317,331,347,349,373,379,389,397,401,419,421,443,461,467,491 %N A059263 Primes p such that x^10 = 2 has no solution mod p. %C A059263 Complement of A049542 relative to A000040. %Y A059263 A000040, A049542. %K A059263 easy,nonn %O A059263 0,1 %A A059263 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 23 2001 %I A059638 %S A059638 3,5,11,13,19,29,37,43,47,53,59,61,67,83,101,107,109,131,139,149,157, %T A059638 163,173,179,181,197,211,227,229,251,269,277,283,293,307,317,331,347, %U A059638 349,373,379,389,397,419,421,443,461,467,491,499,509,523,541,547,557 %N A059638 Primes p such that x^46 = 2 has no solution mod p. %Y A059638 Complement of A049578 relative to A000040. %K A059638 nonn %O A059638 0,1 %A A059638 Klaus Brockhaus (klaus-brockhaus@t-online.de), Feb 02 2001 %I A059265 %S A059265 3,5,11,13,19,29,37,43,53,59,61,67,71,83,101,107,109,113,127,131,139, %T A059265 149,157,163,173,179,181,197,211,227,229,239,251,269,277,281,283,293, %U A059265 307,317,331,337,347,349,373,379,389,397,419,421,443,449,461,463,467 %N A059265 Primes p such that x^14 = 2 has no solution mod p. %Y A059265 Complement of A049546 relative to A000040. %K A059265 easy,nonn %O A059265 0,1 %A A059265 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 23 2001 %I A059314 %S A059314 3,5,11,13,19,29,37,43,53,59,61,67,79,83,101,107,109,131,139,149,157, %T A059314 163,173,179,181,197,211,227,229,251,269,277,283,293,307,313,317,331, %U A059314 347,349,373,379,389,397,419,421,443,461,467,491,499,509,521,523,541 %N A059314 Primes p such that x^26 = 2 has no solution mod p. %Y A059314 Complement of A049558 relative to A000040. %K A059314 nonn %O A059314 0,1 %A A059314 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 25 2001 %I A059336 %S A059336 3,5,11,13,19,29,37,43,53,59,61,67,83,101,103,107,109,131,137,139,149, %T A059336 157,163,173,179,181,197,211,227,229,239,251,269,277,283,293,307,317, %U A059336 331,347,349,373,379,389,397,409,419,421,443,461,467,491,499,509,523 %N A059336 Primes p such that x^34 = 2 has no solution mod p. %Y A059336 Complement of A049566 relative to A000040. %K A059336 nonn %O A059336 0,1 %A A059336 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 26 2001 %I A059350 %S A059350 3,5,11,13,19,29,37,43,53,59,61,67,83,101,107,109,131,139,149,157,163, %T A059350 173,179,181,191,197,211,227,229,251,269,277,283,293,307,317,331,347, %U A059350 349,373,379,389,397,419,421,443,457,461,467,491,499,509,523,541,547 %N A059350 Primes p such that x^38 = 2 has no solution mod p. %Y A059350 Complement of A049570 relative to A000040. %K A059350 nonn %O A059350 0,1 %A A059350 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 27 2001 %I A059644 %S A059644 3,5,11,13,19,29,37,43,53,59,61,67,83,101,107,109,131,139,149,157,163, %T A059644 173,179,181,197,211,227,229,233,251,269,277,283,293,307,317,331,347, %U A059644 349,373,379,389,397,419,421,443,461,467,491,499,509,523,541,547,557 %N A059644 Primes p such that x^58 = 2 has no solution mod p. %Y A059644 Complement of A049590 relative to A000040. %K A059644 nonn %O A059644 0,1 %A A059644 Klaus Brockhaus (klaus-brockhaus@t-online.de), Feb 02 2001 %I A059646 %S A059646 3,5,11,13,19,29,37,43,53,59,61,67,83,101,107,109,131,139,149,157,163, %T A059646 173,179,181,197,211,227,229,251,269,277,283,293,307,311,317,331,347, %U A059646 349,373,379,389,397,419,421,443,461,467,491,499,509,523,541,547,557 %N A059646 Primes p such that x^62 = 2 has no solution mod p. %Y A059646 Complement of A049594 relative to A000040. %K A059646 nonn %O A059646 0,1 %A A059646 Klaus Brockhaus (klaus-brockhaus@t-online.de), Feb 02 2001 %I A003629 M2472 %S A003629 3,5,11,13,19,29,37,43,53,59,61,67,83,101,107,109,131,139,149,157,163, %T A003629 173,179,181,197,211,227,229,251,269,277,283,293,307,317,331,347,349, %U A003629 373,379,389,397,419,421,443,461,467,491,499,509,523,541,547,557,563 %N A003629 Primes == +- 3 (mod 8), or, 2 is not a square mod p. %C A003629 Complement of A038873 relative to A000040. %D A003629 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870. %Y A003629 Cf. A000040, A038873. %K A003629 nonn,easy %O A003629 1,1 %A A003629 njas,mb %I A001122 M2473 N0981 %S A001122 3,5,11,13,19,29,37,53,59,61,67,83,101,107,131,139,149,163,173,179,181, %T A001122 197,211,227,269,293,317,347,349,373,379,389,419,421,443,461,467,491,509, %U A001122 523,541,547,557,563,587,613,619,653,659,661,677,701,709,757,773,787,797 %N A001122 Primes with primitive root 2. %D A001122 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864. %D A001122 R. Gupta and M. R. Murty: A remark on Artin's conjecture, Invent. Math. 78 (1984) 127-230. %D A001122 C. Hooley: On Artin's conjecture, J Reine Angewandte Math., 225 (1967) 209-220. %D A001122 M. Kraitchik, Recherches sur la Th\'{e}orie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 56. %t A001122 << NumberTheoryNumberTheoryFunctions; Prime[ Select[ Range[150], PrimitiveRoot[ Prime[ # ] ] == 2 & ] ] %o A001122 (Pari) forprime(p=3,1000,if(znprimroot(p)==2,print(p))). %Y A001122 Cf. A001123. %K A001122 nonn,easy,nice %O A001122 1,1 %A A001122 njas %I A045407 %S A045407 3,5,11,13,23,31,41,43,53,61,71,73,83,101,103,113,131,151,163,173,181, %T A045407 191,193,211,223,233,241,251,263,271,281,283,293,311,313,331,353,373, %U A045407 383,401,421,431,433,443,461,463 %N A045407 Primes congruent to {0, 1, 3} mod 5. %K A045407 nonn %O A045407 0,1 %A A045407 njas %I A006794 M2474 %S A006794 3,5,11,13,41,89,317,337,991,1873,2053,2377,4093,4297,4583,6569, %T A006794 13033,15877 %N A006794 -1 + product of primes up to p is prime. %C A006794 Or, p such that primorial(p) - 1 is prime. %D A006794 C. K. Caldwell, On The Primality of n!+- 1 and 2\cdot 3 \cdot 5\cdots p+- 1, Math. Comput. 64, 889-890, 1995. %D A006794 H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203. %H A006794 C. K. Caldwell, Primorial Primes %H A006794 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A006794 A057704 gives same sequence in a different way. A057705 gives the actual primes. Cf. A002110, A005234, A014545, A018239. %K A006794 nonn,hard,nice %O A006794 1,1 %A A006794 njas %E A006794 Stated incorrectly in CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 101; corrected in 2nd printing. %E A006794 Corrected by Arlin Anderson (arlin@myself.com), who reports that he and Don Robinson have checked this sequence through about 63000 digits without finding another term (Jul 04 2000). %I A014563 %S A014563 3,5,11,13,41,89,317,337,991,1873,2053,2377,4093,4297,4583,6569,13033, %T A014563 15877 %N A014563 Same as A006794. %C A014563 I have left this entry in place because there may be external links to it. %K A014563 dupe %O A014563 1,1 %I A032457 %S A032457 3,5,11,13,51,53,63,71,103,179,205,267,351,589,719,1453,3703,7935, %T A032457 10851,15151,16013,20319,65475,66083 %N A032457 161*2^n+1 is prime. %H A032457 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A032457 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %H A032457 R. Ballinger and W. Keller, More information %H A032457 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %K A032457 nonn,hard %O A032457 0,1 %A A032457 njas %I A058595 %S A058595 1,3,5,11,13,61,115,653,889,957,997,1013,1421,2329,2675,4795,5755 %N A058595 7*2^n + 5 is prime. %t A058595 Do[ If[ PrimeQ[ 7*2^n + 5 ], Print[ n ] ], {n, 1, 6800} ] %K A058595 nonn %O A058595 1,2 %A A058595 Robert G. Wilson v (rgwv@kspaint.com), Dec 26 2000 %I A024897 %S A024897 3,5,11,15,17,21,27,29,35,39,45,47,53,69,71,75,77,81,83,87,89,95,99,101,113, %T A024897 119,123,131,141,143,147,153,161,165,167,171,183,185,201,203,207,209,213, %U A024897 221,225,245,249,251,255,257,263,279,281,285,287,291,297,299,309,311,315 %N A024897 5*n+4 is prime. %K A024897 nonn %O A024897 1,1 %A A024897 Clark Kimberling (ck6@cedar.evansville.edu) %I A048702 %S A048702 0,1,3,5,11,15,17,21,43,51,55,63,65,73,77,85,171,187,195, %T A048702 211,215,231,239,255,257,273,281,297,301,317,325,341,683, %U A048702 715,731,763,771,803,819,851,855,887,903,935,943,975,991 %N A048702 Binary palindromes of even length divided by 3. a(n) = A048701[ n ]/3 %C A048702 Two unproved formulae which are not based upon first generating a palindrome and then dividing by 3, recursive and more direct: # Here d is 2^(the distance between the most and least significant 1-bit of n): %C A048702 bper3_rec := proc(n) option remember; local d; if(0 = n) then RETURN(0); fi; d := 2^([ log2(n) ]-A007814[ n ]); %C A048702 if(1 = d) then RETURN((2*bper3_rec(n-1))+d); else RETURN(bper3_rec(n-1)+d); fi; end; %C A048702 or more directly (after K. Atanassov's formula for partial sums of A007814): %C A048702 bper3_direct := proc(n) local l,j; l := [ log2(n) ]; RETURN((2/3*((2^(2*l))-1))+1+ sum('(2^(l-j)*floor((n-(2^l)+2^j)/(2^(j+1))))','j'=0..l)); end; %C A048702 Can anybody find an even simpler closed form? See A005187 for inspiration. %Y A048702 A048704 base 4 palindromes of even length divided by 5; A044051 binary palindromes plus one divided by 2 (A006995[ n ]+1)/2. Cf. A000975. %K A048702 nonn %O A048702 0,3 %A A048702 Antti.Karttunen@iki.fi (karttu@megabaud.fi), 7.Mar 1999 %I A003546 %S A003546 1,3,5,11,15,17,25,31,33,41,51,55,75,85,93,123,155,165, %T A003546 187,205,255,275,341,425,451,465,527,561,615,697,775,825, %U A003546 935,1023,1025,1271,1275,1353,1581,1705,2091,2255,2325 %N A003546 Divisors of 2^40 - 1. %K A003546 nonn,fini %O A003546 0,2 %A A003546 njas %I A034169 %S A034169 1,3,5,11,15,21,29,35,39,51,65,95,105,165,231 %N A034169 Disjoint discriminants (one form per genus) of type 2. %D A034169 Borwein and Borwein, Pi and the AGM, page 293. %D A034169 L. E. Dickson, Introduction to the theory of numbers, Dover, NY, 1929. %H A034169 J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx,, Experimental Mathematics, 9 (2000), 153-158 (dvi, ps). %H A034169 Experimental Mathematics, Home Page %Y A034169 Cf. A034168, A034170. %K A034169 nonn,fini,nice %O A034169 1,2 %A A034169 Jonathan Borwein (jborwein@cecm.sfu.ca), choi@cecm.sfu.ca (Stephen Choi) %I A003529 %S A003529 1,3,5,11,15,25,31,33,41,55,75,93,123,155,165,205,275,341, %T A003529 451,465,615,775,825,1023,1025,1271,1353,1705,2255,2325, %U A003529 3075,3813,5115,6355,6765,8525,11275,13981,19065,25575 %N A003529 Divisors of 2^20 - 1. %K A003529 nonn,fini %O A003529 0,2 %A A003529 njas %I A018667 %S A018667 1,3,5,11,15,25,33,55,75,165,275,825 %N A018667 Divisors of 825. %K A018667 nonn,fini,full %O A018667 0,2 %A A018667 njas %I A032673 %S A032673 3,5,11,15,27,29,33,35,41,51,59,63,69,71,89,93,95,101,105,107,117, %T A032673 123,147,155,167,179,183,189,197,201,203,209,219,231,233,245,249,251, %U A032673 261,263,275,281,285,291,303,309,323,329,339,357,365,369,387,389,395 %N A032673 Digit '1' concatenated with a(n) is a lucky number. %Y A032673 Cf. A000959. %K A032673 nonn %O A032673 0,1 %A A032673 Patrick De Geest (pdg@worldofnumbers.com), may 1998. %I A018313 %S A018313 1,3,5,11,15,33,55,165 %N A018313 Divisors of 165. %K A018313 nonn,fini,full %O A018313 0,2 %A A018313 njas %I A006169 M2475 %S A006169 1,3,5,11,16,32,47,84,124,206,299,481,687,1058,1506,2255,3163,4638,6444,9258 %N A006169 Factorization patterns of n. %D A006169 R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82. %K A006169 nonn %O A006169 1,2 %A A006169 njas %I A045408 %S A045408 3,5,11,17,19,29,31,43,47,53,59,61,67,71,73,89,101,103,109,113,127,131, %T A045408 137,151,157,173,179,193,197,199,211,227,229,239,241,257,263,269,271, %U A045408 277,281,283,311,313,337,347 %N A045408 Primes congruent to {1, 3, 4, 5} mod 7. %K A045408 nonn %O A045408 0,1 %A A045408 njas %I A045409 %S A045409 3,5,11,17,19,31,47,53,59,61,67,73,89,101,103,109,131,137,151,157,173, %T A045409 179,193,199,227,229,241,257,263,269,271,277,283,311,313,347,353,367, %U A045409 383,389,397,409,431,439,467,479 %N A045409 Primes congruent to {3, 4, 5} mod 7. %K A045409 nonn %O A045409 0,1 %A A045409 njas %I A050566 %S A050566 3,5,11,17,21,27,81,101,107,327,383,387,941,1665,3515,4007,4727,7137, %T A050566 16257,16833,21243,24239,35435 %N A050566 81*2^n-1 is prime. %H A050566 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A050566 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A050566 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %K A050566 hard,nonn %O A050566 0,1 %A A050566 njas, Dec 29 1999 %I A032382 %S A032382 1,3,5,11,17,21,29,47,85,93,129,151,205,239,257,271,307,351,397,479, %T A032382 553,1317,1631,1737,1859,1917,1999,2353,3477,4331,4337,5809,8707,9649, %U A032382 10421,11117,17425,18343,20791,21035,28331,34249,45029,73237,84839 %N A032382 65*2^n+1 is prime. %H A032382 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A032382 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A032382 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %H A032382 Y. Gallot, Proth.exe: Windows Program for Finding Large Primes %H A032382 R. Ballinger and W. Keller, More information %K A032382 nonn %O A032382 1,2 %A A032382 Jim Buddenhagen (jbuddenh@texas.net) %I A045410 %S A045410 3,5,11,17,23,29,41,47,53,59,71,83,89,101,107,113,131,137,149,167,173, %T A045410 179,191,197,227,233,239,251,257,263,269,281,293,311,317,347,353,359, %U A045410 383,389,401,419,431,443,449,461 %N A045410 Primes congruent to {3, 5} mod 6. %K A045410 nonn %O A045410 0,1 %A A045410 njas %I A063693 %S A063693 3,5,11,17,24,29,41,42,56,59,71,98,101,102,107,137,149,179,191,197,227, %T A063693 230,239,248,264,269,281,294,311,347,419,431,461,468,521,569,599,617, %U A063693 638,641,659,809,821,827,857,881,1014,1016,1019,1031,1049,1061,1078 %N A063693 Solutions to Phi[x+d[x]] = Phi[x]+d[x], where Phi[] = A000010(), d[] = A000005(). %e A063693 Primes and composites among solutions: x = 59, d(58) = 2, Phi(59) = 58, Phi(59+2) = Phi(61) = 60 = Phi(59)+d(59); x = 56, d(56) = 8, Phi(56) = 24, Phi(56+8) = Phi(64) = 32 = Phi(56)+d(56). %Y A063693 A000005, A000010. %K A063693 nonn %O A063693 1,1 %A A063693 Labos E. (labos@ana1.sote.hu), Aug 23 2001 %I A049752 %S A049752 0,3,5,11,17,25,35,49,61,77,99,119,135,163,189,219,247,283,309,343,385, %T A049752 421,463,505,553,603,645,699,751,799,857,919,979,1039 %N A049752 a(n)=T(n,n), array T as in A049747. %K A049752 nonn %O A049752 0,2 %A A049752 Clark Kimberling, ck6@cedar.evansville.edu %I A063700 %S A063700 3,5,11,17,29,41,59,71,101,107,137,147,149,179,191,197,227,239,269,281, %T A063700 311,347,419,431,461,521,569,596,599,617,641,659,809,821,827,857,881, %U A063700 1019,1031,1049,1061,1091,1151,1229,1277,1289,1301,1319,1427,1451,1481 %N A063700 Solutions to Sigma[x+d[x]] = Sigma[x]+d[x], where Sigma[] = A000203(), d[] = A000005(). %e A063700 Primes and composites among solutions: x = 59, d(59) = 2,Sigma(59) = 60,Sigma(59+2) = Sigma(61) = 62 = Sigma(59)+d(59); x = 596, d(596) = 6,Sigma(596) = 1050,Sigma(596+6) = Sigma(602) = 1056 = Sigma(596)+d(596). %Y A063700 A000005, A000203. %K A063700 nonn %O A063700 1,1 %A A063700 Labos E. (labos@ana1.sote.hu), Aug 23 2001 %I A054799 %S A054799 3,5,11,17,29,41,59,71,101,107,137,149,179,191,197,227,239,269,281,311, %T A054799 347,419,431,434,461,521,569,599,617,641,659,809,821,827,857,881,1019, %U A054799 1031,1049,1061,1091,1151,1229,1277,1289,1301,1319,1427,1451,1481,1487 %N A054799 Integers n such that Sigma[n+2]=Sigma[n]+2, Sigma=A000203, sum of divisors of n. %C A054799 Below 1000000 only 3 composite numbers were found:434,8575,8825, So this sequence is different both from A001359 and also from A015818. %D A054799 Sivaramakrishnan, R. (1989): Classical Theory of Arithmetical Functions., M.Dekker Inc., New York, Problem 12 in Chapter V., p. 81. %e A054799 n = 434, divisors = {1, 2, 7, 14, 31, 62, 217, 434}, Sigma[434] = 768, Sigma[436] = 770; n = 8575, divisors = {1, 5, 7, 25, 35, 49, 175, 245, 343, 1225, 1715, 8575}, Sigma[8575] = 12400, Sigma[8577] = 12402; n = 8825, divisors = {1, 5, 25, 353, 1765, 8825}, Sigma[8525] = 10974, Sigma[8527] = 10976 %Y A054799 Cf. A000203, A001359, A015818, A050507. %K A054799 nonn %O A054799 1,1 %A A054799 Labos E. (labos@ana1.sote.hu), May 22 2000 %I A015818 %S A015818 3,5,11,17,29,41,59,71,101,107,137,149,179,191,197,227,239,269,281,311, %T A015818 347,419,431,461,521,569,599,617,641,659,809,821,827,857,881,1019,1031, %U A015818 1049,1061,1091,1151,1229,1277,1289,1301,1319,1427,1451,1481,1487 %N A015818 phi(n + 2) = sigma(n). %K A015818 nonn,easy %O A015818 0,1 %A A015818 Robert G. Wilson v (rgwv@kspaint.com) %E A015818 Corrected and extended by Jud McCranie (jud.mccranie@mindspring.com), Jan 03 2001, who observes that for terms < 4293000000 this coincides with A001359, and asks if they are the same sequence. %I A001359 M2476 N0982 %S A001359 3,5,11,17,29,41,59,71,101,107,137,149,179,191,197,227,239,269,281,311, %T A001359 347,419,431,461,521,569,599,617,641,659,809,821,827,857,881,1019,1031, %U A001359 1049,1061,1091,1151,1229,1277,1289,1301,1319,1427,1451,1481,1487,1607 %N A001359 Lesser of twin primes. %D A001359 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870. %D A001359 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 6. %D A001359 T. R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204. %H A001359 C. K. Caldwell, Twin Primes %H A001359 C. K. Caldwell, Largest known twin primes %H A001359 Caldwell's prime pages %H A001359 Thomas R. Nicely's home page, which has extensive tables. %H A001359 C. K. Caldwell, Twin primes %H A001359 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A001359 Cf. A006512 (greater of twin primes), A014574, A015818. %K A001359 nonn,nice,easy %O A001359 1,1 %A A001359 njas %I A023218 %S A023218 3,5,11,17,29,47,53,71,83,89,101,113,131,167,251,257,263,281,311,389,419, %T A023218 461,467,479,491,509,521,557,563,587,593,599,617,641,659,677,743,797,809, %U A023218 827,857,881,929,977,983,1019,1061,1103,1187,1217,1259,1277,1289,1319 %N A023218 n and 5n + 4 both prime. %K A023218 nonn %O A023218 1,1 %A A023218 dww %I A006450 M2477 %S A006450 3,5,11,17,31,41,59,67,83,109,127,157,179,191,211,241,277,283,331,353, %T A006450 367,401,431,461,509,547,563,587,599,617,709,739,773,797,859,877,919, %U A006450 967,991,1031,1063,1087,1153,1171,1201,1217,1297,1409,1433,1447,1471 %N A006450 Primes with prime subscripts. %D A006450 R. E. Dressler and S. T. Parker, Primes with a prime subscript, J. ACM, 22 (1975), 380-381. %H A006450 N. Fernandez, An order of primeness, F(p) %H A006450 N. Fernandez, More terms of this and other sequences related to A049076. %t A006450 Table[ Prime[ Prime[ n ] ],{n,1,51} ] %Y A006450 Primes for which A049076 > 1. Cf. A038580, A049076-A049081. %K A006450 nonn,easy,nice %O A006450 1,1 %A A006450 jos,sm %I A006170 M2478 %S A006170 1,3,5,11,17,33,50,89,135,223,332,531,776,1194,1730,2591,3700,5429,7660,11035 %N A006170 Factorization patterns of n. %D A006170 R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82. %K A006170 nonn %O A006170 1,2 %A A006170 njas %I A006171 M2479 %S A006171 1,3,5,11,17,34,52,94,145,244,370,603,899,1410,2087,3186,4650,6959, %T A006171 10040,14750,21077,30479,43120,61574,86308,121785,169336,236475,326201,451402 %N A006171 Factorization patterns of n. %D A006171 R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82. %H A006171 N. J. A. Sloane, Transforms %F A006171 Euler transform of tau(n), tau(n)=the number of divisors of n, cf. A000005. G.f.: Product_{k=1..infinity} (1 - x^k)^(-tau(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*tau(d), cf. A060640. - Vladeta Jovovic (vladeta@Eunet.yu), Apr 21 2001 %Y A006171 Cf. A000005, A060640, A061255, A061256, A001970, A061257. %K A006171 nonn %O A006171 1,2 %A A006171 njas %I A060647 %S A060647 1,3,5,11,17,35,53,107,161,323,485,971,1457,2915,4373,8747,13121,26243, %T A060647 39365,78731,118097,236195,354293,708587,1062881,2125763,3188645, %U A060647 6377291,9565937,19131875,28697813,57395627,86093441,172186883 %N A060647 Number of alpha-beta evaluations in a tree of depth n and branching factor b=3. %D A060647 P. H. Winston, Artificial Intelligence, (1977) 115-122, alpha-beta technique. %F A060647 a(2n) = 2 * (3^n) -1, a(2n+1) = 3^n +3^(n+1) -1. %F A060647 Formula for b branches: a(2n)=2*(b^n)-1, a(2n+1)=b^n +b^(n+1) -1 %e A060647 a(2n+1) = 2*a(2n) +1, a(15) = a(2*7+1) = 2*a(14) +1 = 2*4373 +1 = 8747 %p A060647 A060647:=proc(n,b) option remember: if n mod 2 = 0 then RETURN(2*b^(n/2)-1) else RETURN(b^((n-1)/2) +b^((n+1)/2)-1) fi: end: for n from 0 to 60 do printf(%d,,A060647(n,3)) od: %Y A060647 for b=2 see A052955. %K A060647 easy,nonn %O A060647 0,2 %A A060647 Frank Ellermann (Frank.Ellermann@t-online.de), Apr 17 2001 %E A060647 More terms from James A. Sellers (sellersj@math.psu.edu), Apr 19 2001 %I A007455 M2480 %S A007455 1,1,3,5,11,17,39,61,139,217,495,773,1763,2753,6279,9805,22363,34921, %T A007455 79647,124373,283667,442961,1010295,1577629,3598219,5618809,12815247, %U A007455 20011685,45642179,71272673,162557031,253841389,578955451,904069513 %N A007455 Subsequences of [ 1,...,n ] in which each odd number has an even neighbor. %D A007455 R. K. Guy, Moser, William O.J.: Numbers of subsequences without isolated odd members. Fibonacci Quarterly, 34, No.2, 152-155 (1996). %F A007455 a(n) = 3*a(n-2) + 2*a(n-4). %Y A007455 Cf. A007481, A007482, A007484. %K A007455 nonn,easy,nice %O A007455 0,3 %A A007455 njas,mb %I A034729 %S A034729 1,3,5,11,17,39,65,139,261,531,1025,2095,4097,8259,16405,32907,65537, %T A034729 131367,262145,524827,1048645,2098179,4194305,8390831,16777233, %U A034729 33558531,67109125,134225995,268435457,536887863,1073741825,2147516555 %N A034729 Dirichlet convolution of b_n=1 with c_n=2^(n-1). %K A034729 nonn %O A034729 1,2 %A A034729 Erich Friedman (erich.friedman@stetson.edu) %I A040176 %S A040176 1,3,5,11,17,41,83,137,257,641,1097,2329,4369,10537,17477,35209,65537, %T A040176 140417,281929,557057,1114129,2384897,4227137,8978569,16843009, %U A040176 35946497,71304257,143163649,286331153,541073537,1086374209,2281701377,4295098369 %N A040176 Smallest number which when iterated n times under Euler's totient function phi(n) [A000010] gives 1. %C A040176 Smallest m such that phi(phi(..(m)))=1 with n iterations. %C A040176 a(n) is smallest number in the class k(n) which groups families of integers which take the same number of iterations of the totient function to evolve to 1. The maximum is 2*3^(n-1). %D A040176 R. K. Guy,; Unsolved Problems in Number Theory, 2nd Ed. New York: Springer-Verlag, p. 97, 1994, Section B41. %e A040176 phi(phi(3)))=1, 3 is the smallest number requiring 2 iterations to reach 1, so a(2)=3. %Y A040176 Cf. A000010, A003434, A049108, A007755 (another version of this sequence; A040176 is the correct one). %K A040176 nice,nonn,easy %O A040176 1,2 %A A040176 James S. Cronen (cronej@rpi.edu) %E A040176 More terms from Jud McCranie (jud.mccranie@mindspring.com), Jan 03 2000 %I A019386 %S A019386 3,5,11,17,43,47,71,73,83,89,103,109,137,149,157,173,179,191,241,251,293, %T A019386 311,313,317,347,349,353,359,373,401,409,431,433,439,449,457,479,491,503, %U A019386 541,569,587,599,607,619,643,653,701,727,733,739,751,787,809,827,839,853 %N A019386 Primes with primitive root 62. %K A019386 nonn %O A019386 1,1 %A A019386 dww %I A057652 %S A057652 3,5,11,17,647 %N A057652 n such that positive values of n-2^k are all lucky numbers (k>0). %Y A057652 Cf. A000959, A039669. %K A057652 more,nonn %O A057652 0,1 %A A057652 Naohiro Nomoto (6284968128@geocities.co.jp), Oct 14 2000 %I A023597 %S A023597 1,3,5,11,18,30,51,84,137,222,362,587,952,1541,2494,4038,6535,10576, %T A023597 17114,27692,44807,72502,117312,189817,307132,496952,804086,1301039, %U A023597 2105128,3406170,5511301,8917474 %N A023597 Convolution of A023532 and Lucas numbers. %K A023597 nonn %O A023597 1,2 %A A023597 Clark Kimberling (ck6@cedar.evansville.edu) %I A045957 %S A045957 3,5,11,19,21,35,43,51,53,69,99,101,115,117,131,139,149,171,197,213,229, %T A045957 245,261,299,309,325,373,387,389,419,435,437,517,523,533,547,587,597,629, %U A045957 643,645,707,709,715,739,741,779,869,883,885,915,963,965,995,997,1091 %N A045957 Twin even-lucky-numbers: middle terms. %Y A045957 Cf. A045954. %K A045957 base,nonn %O A045957 1,1 %A A045957 Felice Russo (felice.russo@katamail.com) %I A023233 %S A023233 3,5,11,19,29,31,43,53,71,73,101,103,109,113,131,151,173,179,191,211,229, %T A023233 233,239,269,271,281,283,311,313,373,379,383,431,443,491,499,509,521,541, %U A023233 599,613,619,653,691,701,719,733,739,743,751,773,809,883,919,929,971,983 %N A023233 n and 9n + 2 both prime. %K A023233 nonn %O A023233 1,1 %A A023233 dww %I A048161 %S A048161 3,5,11,19,29,59,61,71,79,101,131,139,181,199,271,349,379,409,449, %T A048161 461,521,569,571,631,641,661,739,751,821,881,929,991,1031,1039,1051, %U A048161 1069,1091,1129,1151,1171,1181,1361,1439,1459,1489,1499,1531,1709 %N A048161 Each prime is a leg of an integral right triangle whose hypotenuse is also prime. %C A048161 It is conjectured that there are an infinite number of such triangles. %H A048161 H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3. %F A048161 p is prime such that q=(p*p+1)/2 is also prime. %e A048161 For p(1)=3, the right triangle 3, 4, 5 is the smallest where 5=(3*3+1)/2. For p(10)=101, the right triangle is 101, 5100, 5101 where 5101=(101*101+1)/2. %K A048161 nonn,nice %O A048161 1,1 %A A048161 Harvey Dubner (hdubner1@compuserve.com) %E A048161 More terms from dww %I A051642 %S A051642 3,5,11,19,29,101,349,521,569,571,661,1091,1489,2269,2341,2549,2659, %T A051642 2729,2731,4049 %N A051642 Values of A (the short leg) of a Pythagorean triangle with A and C (the hypotenuse) both prime and part of a twin prime. %K A051642 nonn,easy,more %O A051642 0,1 %A A051642 stuart m. ellerstein (ellerstein@aol.com) %I A007671 M2481 %S A007671 1,3,5,11,19,29,157,163,283,379,997,10141,14699,77291,85237,106693,203789 %N A007671 2^n + 2^(n + 1)/2 + 1 is prime. %D A007671 J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements. %H A007671 S. S. Wagstaff, Jr., The Cunningham Project %Y A007671 Cf. A057429. %K A007671 hard,nonn %O A007671 1,2 %A A007671 njas, Robert G. Wilson v (rgwv@kspaint.com) %E A007671 More terms from Robert G. Wilson v (RGWv@kspaint.com), Sep 07 2000 %E A007671 203789 found and proved prime by Mike Oakes (mikeoakes2@aol.com), on 28 Sep 2000. %I A045691 %S A045691 0,1,1,3,5,11,19,41,77,159,307,625,1231,2481,4921,9883,19689,39455, %T A045691 78751 %N A045691 Binary words of length n (beginning 0) with autocorrelation function 2^(n-1)+1. %F A045691 a[ 2n-1 ] = 2 a[ 2n-2 ] - a[ n ] for n >= 2; a[ 2n ] = 2 a[ 2n-1 ] + a[ n ] for n >= 2 %K A045691 nonn %O A045691 1,4 %A A045691 TORSTEN.SILLKE@LHSYSTEMS.COM %I A045961 %S A045961 3,5,11,19,43,53,101,131,139,149,197,229,373,389,419,523,547,587,643,709, %T A045961 739,883,997,1091,1093,1187,1483,1621,1931,1973,2099,2243,2347,2357,2411, %U A045961 2549,2677,2731,2741,2803,2963,3011,3203,3307,3331,3461,3467,3541,3733 %N A045961 Twin A045954's (middle terms) that are primes. %Y A045961 Cf. A045954. %K A045961 nonn %O A045961 1,1 %A A045961 Felice Russo (felice.russo@katamail.com) %E A045961 More terms from dww %I A061068 %S A061068 3,5,11,19,79,101,113,127,163,173,223,271,383,419,431,503,571,599,619, %T A061068 641,659,673,683,701,733,757,827,863,971,1013,1033,1087,1193,1249,1423, %U A061068 1433,1453,1483,1579,1621,1667,1723,2003,2113,2179,2287,2381,2459,2467 %N A061068 Primes of special form: sum of a smaller prime and its prime subscript: a(n)=w+p(w). %e A061068 5th term here is 69=61+18=p(18)+18. %Y A061068 Cf. A061067. %K A061068 nonn %O A061068 0,1 %A A061068 Labos E. (labos@ana1.sote.hu), May 28 2001 %I A058932 %S A058932 0,0,0,0,0,0,1,1,3,5,11,20 %N A058932 Unlabeled claw-free cubic graphs with 2n nodes and connectivity 1. %D A058932 G.-B. Chae (chaegabb@pilot.msu.edu), E. M. Palmer, and R. W. Robinson, Computing the number of Claw-free Cubic Graphs with given Connectivity, preprint, 2001. %H A058932 Home page of Chae %K A058932 nonn %O A058932 1,9 %A A058932 njas, Jan 12 2001 %I A057460 %S A057460 1,3,5,11,21,29,35,93,123,333 %N A057460 x^n + x^2 + 1 is irreducible over GF(2). %t A057460 Do[ If[ ToString[ Factor[ x^n + x^2 + 1, Modulus -> 2 ] ] == ToString[ x^n + x^2 + 1 ], Print[ n ] ], {n, 0, 750} ] %Y A057460 Cf. A002475. %K A057460 nonn %O A057460 0,2 %A A057460 Robert G. Wilson v (rgwv@kspaint.com), Sep 27 2000 %I A045515 %S A045515 0,1,3,5,11,21,35,63,97,153,229,351,474,714 %N A045515 Conjugacy classes of elements of order n in 2.E_7(C). %D A045515 Cohen, Arjeh M.; Griess, Robert L., Jr.; On finite simple subgroups of the complex Lie group of type $E\sb 8$. The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 367-405, Proc. Sympos. Pure Math., 47, Part 2, Amer. Math. Soc., Providence, RI, 1987. %K A045515 nonn %O A045515 0,3 %A A045515 njas %I A001045 M2482 N0983 %S A001045 0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691, %T A001045 87381,174763,349525,699051,1398101,2796203,5592405,11184811,22369621, %U A001045 44739243,89478485,178956971,357913941,715827883,1431655765,2863311531 %N A001045 Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2). %C A001045 Number of ways to tile a 3x(n-1) rectangle with 1 X 1 and 2 X 2 square tiles. %C A001045 Also the number of ways to tie a necktie using n+2 turns. So three turns make an "oriental", four make a "four in hand", and for 5 turns there are 3 methods: "Kelvin", "Nicky" and "Pratt". The formula also arises from a special random walk on a triangular grid with side conditions (see Fink and Mao, 1999). - arne.ring@epost.de, 18 Mar 2001 %C A001045 Also the number of compositions of n+1 ending with an odd part (a(2)=3 because 3, 21, 111 are the only compositions of 3 ending with an odd part). Also the number of compositions of n+2 ending with an even part (a(2)=3 because 4, 22, 112 are the only compositions of 4 ending with an even part). - Emeric Deutsch (deutsch@duke.poly.edu), May 08 2001 %D A001045 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550. %D A001045 D. E. Daykin, D. J. Kleitman and D. B. West, The number of meets between two subsets of a lattice, J. Combin. Theory, A 26 (1979), 135-156. %D A001045 Th. Fink and Y. Mao. The 85 ways to tie a tie, Fourth Estate, London, 1999; Die 85 Methoden eine Krawatte zu binden. Hoffmann und Kampe, Hamburg, 1999. %D A001045 S. L. Levine, Suppose more rabbits are born, Fib. Quart., 26 (1988), 306-311. %D A001045 G. Myerson and A. J. van der Poorten, Some problems concerning recurrence sequences, Amer. Math. Monthly, 102 (1995), 698-705. %D A001045 Two-Year College Math. Jnl., 28 (1997), p. 76. %D A001045 G. B. M. Zerr, Problem 64, American Mathematical Monthly, vol. 3, no. 12, 1896 (p. 311). %H A001045 D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.3 %H A001045 S. Heubach, Tiling an m X n area with squares of size up to k X k (m <=5), Congressus Numerantium 140 (1999), pp. 43-64. %H A001045 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 142 %H A001045 E. W. Weisstein, Link to a section of The World of Mathematics. %F A001045 a(n) = 2^n - a(n-1). a(n) = 2*a(n-1) - (-1)^n = {2^n - (-1)^n}/3. %e A001045 a(2) = 3 because the tiling of the 3x2 rectangle has either only 1 X 1 tiles, or one 2 X 2 tile in one of two positions (together with 2 1 X 1 tiles) %Y A001045 Partial sums of this sequence give A000975, where there are additional comments from B. E. Williams and Bill Blewett on the tie problem. Cf. A049883. %K A001045 nonn,nice,easy %O A001045 0,4 %A A001045 njas %E A001045 More terms from James A. Sellers (sellersj@math.psu.edu), Dec 23 1999. Zerr reference from Len Smiley (smiley@math.uaa.alaska.edu), May 21 2001. %I A007873 %S A007873 1,1,3,5,11,21,45,85,179,341,717,1365,2867,5461,11475,21845,45875, %T A007873 87381,183597,349525,734003,1398101,2937555,5592405,11744051, %U A007873 22369621,47000877,89478485,187904819 %N A007873 Indices of last windows of trapezoidal maps. %D A007873 P H Borcherds and G P McCauley, The digital tent map and trapezoidal map, in "Chaos, Solitons and Fractals", Vol 3, pp. 451-66 (1993) %K A007873 nonn %O A007873 1,3 %A A007873 Peter H. Borcherds [ phb@phymat.bham.ac.uk ] %I A004039 M2483 %S A004039 1,1,1,3,5,11,22,26,42,70,112 %N A004039 The coding-theoretic function A(n,6,6). %D A004039 A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380. %D A004039 CRC Handbook of Combinatorial Designs, 1996, p. 411. %H A004039 E. M. Rains and N. J. A. Sloane, A(n,d,w) tables %H A004039 Index entries for sequences related to A(n,d,w) %K A004039 nonn,hard %O A004039 6,4 %A A004039 njas %I A005830 %S A005830 1,3,5,11,22,46,96,201,421,882,1848,3872,8113,16999,35617, %T A005830 74628,156367,327635,686491,1438400,3013871,6314945,13231666, %U A005830 27724230,58090412,121716491,255031831,534366661,1119655250 %N A005830 a(n) = [ tau*a(n-1) ] + a(n-2). %K A005830 nonn %O A005830 0,2 %A A005830 njas %I A007008 M2484 %S A007008 0,1,1,3,5,11,22,47,93,193,386,793,1586,3238,6476,13167,26333,53381, %T A007008 106762,215955 %N A007008 Chvatal conjecture for radius of graph of maximal intersecting sets. %D A007008 A. Meyerowitz, Maximal intersecting families, European J. Combin. 16 (1995), no. 5, 491-501. %H A007008 D. E. Loeb and A. Meyerowitz, The maximal intersecting family of sets graph, in H. Barcelo and G. Kalai, editors, Proceedings of the Conference on Jerusalem Combinatorics 1993. AMS series Contemporary Mathematics, 1994. %K A007008 nonn %O A007008 1,4 %A A007008 loeb@delanet.com (Daniel LOEB) %I A018110 %S A018110 1,3,5,11,22,48,104,224,484,1049,2271,4917,10648,23061, %T A018110 49944,108165,234256,507337,1098759,2379621,5153632,11161411, %U A018110 24172677,52351655,113379904,245551028,531798889,1151736408 %N A018110 Powers of fourth root of 22 rounded up. %K A018110 nonn %O A018110 0,2 %A A018110 njas %I A015915 %S A015915 3,5,11,23,27,29,53,59,71,89,101,131,149,173,191,233,263,269,359, %T A015915 389,401,431,449,479,491,563,569,593,599,653,683,701,719,743,761, %U A015915 821,911,929,983,1013,1031,1061,1109,1163,1193,1223,1229,1283,1289 %N A015915 sigma(n) + 8 = sigma(n + 8). %C A015915 Different from A023202. Below 1000000 four composites were found [27, 1615, 1885, 218984] satisfying the "Sigma[x+8]=Sigma[x]+8" relation, together with more than 8000 primes - Labos E. (labos@ana1.sote.hu), May 23 2000 %e A015915 sigma(27)+8=48=sigma(27+8), so 27 is in the sequence. %Y A015915 Cf. A015913-A015917, A023200-A023203, A046133, A001359, A054799. %Y A015915 Composite solutions are in A059118. %K A015915 nonn %O A015915 0,1 %A A015915 Robert G. Wilson v (rgwv@kspaint.com) %I A023202 %S A023202 3,5,11,23,29,53,59,71,89,101,131,149,173,191,233,263,269,359,389,401, %T A023202 431,449,479,491,563,569,593,599,653,683,701,719,743,761,821,911,929,983, %U A023202 1013,1031,1061,1109,1163,1193,1223,1229,1283,1289,1319,1373,1439,1451 %N A023202 n and n + 8 both prime. %H A023202 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A023202 Disjoint union of A007530, A031926, A049437, A049438. Cf. also A046134, A049436, A046138, A015915. %K A023202 nonn %O A023202 1,1 %A A023202 dww %I A049436 %S A049436 3,5,11,23,29,53,59,71,101,131,149,173,191,233,263,269,431,563,569,593, %T A049436 599,653,821,1013,1031,1061,1223,1229,1283,1289,1319,1451,1481,1601, %U A049436 1613,1619,1871,2081,2129,2333,2339,2381,2543,2549,2711,2789,2963,3251 %N A049436 p, p+8 and either p+2 or p+6 or both are all primes. %e A049436 3 is here because 5, 7 and 11 are primes; 5 is here because 7, 11 and 13 are primes. %Y A049436 A031926, A023202, A049437, A049438, A046134, A046138, A007530. %K A049436 nonn %O A049436 1,1 %A A049436 Labos E. (labos@ana1.sote.hu) %I A056874 %S A056874 3,5,11,23,31,37,47,53,59,67,71,89,97,103,113,137,157,163,179,181, %T A056874 191,199,223,229,251,257,269,311,313,317,331,353,367,379,383,389, %U A056874 397,401,419,421,433,443,449,463,467,487,499,509,521,577,587,599 %N A056874 Primes of form (x^2+11*y^2)/4. %p A056874 a:=[]; for x from 0 to 80 do for y from 1 to 26 do p:=(x^2+11*y^2)/4; if whattype(p) = integer then if isprime(p) then a:=[op(a),[p,x,y]]; fi; fi; od: od: writeto(trans); for i from 1 to 158 do lprint(a[i]); od: # then sort the list in "trans" %Y A056874 Cf. A002346 and A002347 for values of x and y. %K A056874 nonn %O A056874 1,1 %A A056874 njas, Sep 02 2000 %I A023223 %S A023223 3,5,11,23,47,53,71,101,107,131,167,173,197,251,257,293,311,317,353,383, %T A023223 431,461,467,563,587,593,683,701,773,797,821,827,863,887,911,953,977,983, %U A023223 1031,1091,1097,1103,1151,1181,1187,1193,1217,1223,1277,1301,1307,1373 %N A023223 n and 7n + 2 both prime. %K A023223 nonn %O A023223 1,1 %A A023223 dww %I A032803 %S A032803 1,1,3,5,11,23,47,99,203,423,877,1819,3777,7831,16253,33715, %T A032803 69953,145137,301113,624745,1296165,2689221,5579425,11575849, %U A032803 24016893,49828757,103381739,214490133,445011179,923282285 %N A032803 Expansion of sum( q^i*theta_3^i, i=0..inf). %K A032803 nonn %O A032803 0,3 %A A032803 njas %I A030494 %S A030494 1,3,5,11,23,47,119,239,719,1439,5039,10079,40319,80639,362879,725759, %T A030494 3628799,7257599,39916799,79833599,479001599,958003199,6227020799, %U A030494 12454041599,87178291199,174356582399,1307674367999,2615348735999 %N A030494 If n is even, 2(n/2+1)!-1; if n is odd, ((n+1)/2+1)!-1. %Y A030494 s(n)=s(n+1), where s=A030298. %K A030494 nonn,easy %O A030494 1,2 %A A030494 Clark Kimberling, ck6@cedar.evansville.edu %E A030494 Better description and more terms from Erich Friedman (erich.friedman@stetson.edu). %I A027763 %S A027763 3,5,11,23,47,283,1699,611641 %N A027763 Let b(n) denote a tower of n 2s: 2^2^2^2; a(n) is smallest k such that b(n) is not congruent to b(n-1) mod k. %D A027763 Stan Wagon (WAGON@macalester.edu), posting to Problem of the Week mailing list, Dec 1997. %K A027763 nonn %O A027763 2,1 %A A027763 rkg@cpsc.ucalgary.ca (Richard Guy) %I A018113 %S A018113 1,3,5,11,23,51,111,242,529,1159,2537,5556,12167,26645, %T A018113 58351,127785,279841,612835,1342071,2939052,6436343,14095197, %U A018113 30867617,67598189,148035889,324189523,709955184,1554758333 %N A018113 Powers of fourth root of 23 rounded up. %K A018113 nonn %O A018113 0,2 %A A018113 njas %I A037446 %S A037446 0,0,1,1,3,5,11,23,52,120,286,702,1779,4636,12398,33981,95303,273134, %T A037446 798957,2382777,7238273,22376763,70343548,224698356,728843771, %U A037446 2399172217,8010042859,27109839313,92966452913,322876110331 %N A037446 Floor of (n/e)^(n/e). %o A037446 (PARI.2.0.17) v=[];for(n=1,50,v=concat(v,floor((n/exp(1))^(n/exp(1)))));v %K A037446 easy,huge,nonn %O A037446 1,5 %A A037446 Jason Earls (jcearls@kskc.net), Jun 30 2001 %I A051439 %S A051439 3,5,11,23,59,137,313,727,1621,3673,8167,17881,38891,84047,180511, %T A051439 386117,821647,1742539,3681149,7754081,16290073,34136059,71378603, %U A051439 148948141,310248251,645155227,1339484207,2777105131,5750079077 %N A051439 Prime(2^n+1). %Y A051439 Cf. A000040, A033844, A018249, A051438, A051440. %K A051439 nonn %O A051439 0,1 %A A051439 njas %E A051439 More terms from Michael Lugo (mlugo@thelabelguy.com), Dec 22 1999 %I A018116 %S A018116 1,3,5,11,24,54,118,261,576,1275,2822,6246,13824,30598, %T A018116 67724,149897,331776,734342,1625364,3597522,7962624,17624185, %U A018116 39008732,86340517,191102976,422980417,936209560,2072172385 %N A018116 Powers of fourth root of 24 rounded up. %K A018116 nonn %O A018116 0,2 %A A018116 njas %I A018008 %S A018008 1,3,5,11,25,55,121,270,599,1331,2961,6584,14641,32562, %T A018008 72416,161051,358175,796573,1771561,3939917,8762296,19487171, %U A018008 43339081,96385252,214358881,476729884,1060237770,2357947691 %N A018008 Powers of cube root of 11 rounded up. %K A018008 nonn %O A018008 0,2 %A A018008 njas %I A027050 %S A027050 1,3,5,11,25,59,145,367,949,2495,6645,17883,48541,132711,365073, %T A027050 1009647,2805365,7827167,21918997,61584891,173550677,490408623, %U A027050 1389206065,3944231887,11221911849,31989733339,91354992405,261322661051 %N A027050 a(n) = T(n,2n-1), T given by A027036. %K A027050 nonn %O A027050 1,2 %A A027050 Clark Kimberling, ck6@cedar.evansville.edu %I A032364 %S A032364 1,3,5,11,27,43,57,75,77,93,103,143,185,231,245,391,1053,1175,2027, %T A032364 3627,4727,5443,7927,8533,9067,14185,25723,88117,96947 %N A032364 29*2^n+1 is prime. %H A032364 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A032364 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A032364 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %H A032364 Y. Gallot, Proth.exe: Windows Program for Finding Large Primes %H A032364 R. Ballinger and W. Keller, More information %K A032364 nonn,hard %O A032364 1,2 %A A032364 Jim Buddenhagen (jbuddenh@texas.net) %I A045536 %S A045536 3,5,11,29,41,179,191,239,281,419,431,641,659,809,1019,1031,1049,1229, %T A045536 1289,1451,1481,1931,2129,2141,2339,2549,2969,3299,3329,3359,3389,3539, %U A045536 3821,3851,4019,4271,4481,5231,5279,5441,5501,5639,5741,5849,6131 %N A045536 Twin primes which are also Sophie Germain primes. %Y A045536 Cf. A005384. %K A045536 nonn %O A045536 0,1 %A A045536 Thomas Kellar (tkellar@fsp.com) %E A045536 Corrected by Jud McCranie (jud.mccranie@mindspring.com), Dec 30 2000 %I A019338 %S A019338 3,5,11,29,53,59,83,101,107,131,149,173,179,197,227,269,293,317,347,389, %T A019338 419,443,461,467,491,509,557,563,587,653,659,677,701,773,797,821,827,941, %U A019338 947,1019,1061,1091,1109,1187,1229,1259,1277,1283,1301,1307,1373,1427 %N A019338 Primes with primitive root 8. %K A019338 nonn %O A019338 1,1 %A A019338 dww %I A046134 %S A046134 3,5,11,29,59,71,101,149,191,269,431,569,599,821,1031,1061,1229,1289, %T A046134 1319,1451,1481,1619,1871,2081,2129,2339,2381,2549,2711,2789,3251, %U A046134 3299,3461,3539,4019,4049,4091,4649,4721,5099,5441,5519,5639,5651 %N A046134 p, p+2, and p+8 are prime. %H A046134 E. W. Weisstein, Link to a section of The World of Mathematics. %K A046134 nonn %O A046134 1,1 %A A046134 Eric W. Weisstein (eric@weisstein.com) %I A057735 %S A057735 3,5,11,29,83,6563,59051,4782971,14348909,282429536483,2541865828331, %T A057735 150094635296999123,1144561273430837494885949696429, %U A057735 57264168970223481226273458862846808078011946891 %N A057735 Primes of the form 3^n + 2. %K A057735 nonn %O A057735 0,1 %A A057735 G. L. Honaker, Jr. (curios@bvub.com), Oct 29 2000 %I A000101 M2485 N0984 %S A000101 3,5,11,29,97,127,541,907,1151,1361,9587,15727,19661,31469,156007,360749, %T A000101 370373,492227,1349651,1357333,2010881,4652507,17051887,20831533, %U A000101 47326913,122164969,189695893,191913031,387096383,436273291,1294268779 %N A000101 Increasing gaps between primes (upper end). %D A000101 B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133. %D A000101 D. Shanks, On maximal gaps between successive primes, Math. Comp., 18 (1964), 646-651. %D A000101 J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224. %Y A000101 Cf. A002386, A005250. %K A000101 nonn,nice %O A000101 1,1 %A A000101 njas %I A037152 %S A037152 3,5,11,29,127,727,5051,40343,362897,3628811,39916817,479001629, %T A037152 6227020867,87178291219,1307674368043,20922789888023,355687428096031, %U A037152 6402373705728037,121645100408832089 %N A037152 Smallest prime > n!+1. %K A037152 nonn %O A037152 1,1 %A A037152 Jud McCranie (jud.mccranie@mindspring.com) %I A048235 %S A048235 3,5,11,31,76,136,220,324,459,639,870,1146,1523,1971,2481,3073,3770, %T A048235 4544,5399,6339,7347,8513,9801,11169,12669,14307,16062,17994,20082, %U A048235 22302,24627,27059,29699,32453,35358,38454,41673,45207 %N A048235 a(n)=SUM{T(n,j): j=1,2,...,n}, array T given by A048225. %K A048235 nonn %O A048235 1,1 %A A048235 Clark Kimberling, ck6@cedar.evansville.edu %I A063499 %S A063499 3,5,11,31,131,733,362903,39916831,355687428096059,6402373705728061, %T A063499 15511210043330985984000097,8222838654177922817725562880000127, %U A063499 815915283247897734345611269596115894272000000173 %N A063499 Primes of form p(n) + n!, where p(n) is the n-th prime. %o A063499 (PARI) for(n=1,70,x=prime(n)+n!;if(isprime(x),print(x))) %K A063499 easy,huge,nonn %O A063499 1,1 %A A063499 Jason Earls (jcearls@kskc.net), Jul 30 2001 %I A060881 %S A060881 3,5,11,37,221,2323,30047,510529,9699713,223092899,6469693261, %T A060881 200560490167,7420738134851,304250263527253,13082761331670077, %U A060881 614889782588491463,32589158477190044789,1922760350154212639131 %N A060881 n-th primorial (A002110) + next prime. %H A060881 Hisanori Mishima, Factorizations of many number sequences %H A060881 Hisanori Mishima, Factorizations of many number sequences %e A060881 a(2) = 2*3 + 5 = 11. %Y A060881 Cf. A002110, A060882. %K A060881 nonn %O A060881 0,1 %A A060881 njas, May 05 2001 %I A035345 %S A035345 3,5,11,37,223,2333,30047,510529,9699713,223092907,6469693291, %T A035345 200560490197,7420738134871,304250263527281,13082761331670077, %U A035345 614889782588491517,32589158477190044789,1922760350154212639131 %N A035345 Smallest prime > p(1)p(2)...p(n)+1. %D A035345 S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210. %e A035345 Next prime after 2*3*5 + 1 = 31 is 37, so a(3)=37. %Y A035345 Cf. A002110, A005235, A006862. %K A035345 nonn %O A035345 0,1 %A A035345 njas %I A055511 %S A055511 3,5,11,41,89,317,337,991,1873,2053,2377,4093,4297,4583,6569,13033, %T A055511 15877 %N A055511 Erroneous version of A014563. %K A055511 dead %O A055511 1,1 %I A049883 %S A049883 3,5,11,43,683,2731,43691,174763,2796203,715827883,2932031007403, %T A049883 768614336404564651,201487636602438195784363, %U A049883 845100400152152934331135470251,56713727820156410577229101238628035243 %N A049883 Primes in the Jacobsthal sequence (A001045). %K A049883 huge,nonn %O A049883 1,1 %A A049883 Judson D. Neer (judson@poboxes.com) %I A059242 %S A059242 0,1,3,5,11,47,53,141,143 %N A059242 2^n+5 is prime (using Maple's probabilistic isprime). %e A059242 2^3+5 = 13 is prime but 2^4+5 = 21 is not. %K A059242 nonn %O A059242 0,3 %A A059242 Tony Davie (ad@dcs.st-and.ac.uk), Jan 21 2001 %I A004203 %S A004203 1,3,5,11,47,79,275,839,1877 %N A004203 54*10^n + 1 is prime. %D A004203 Computed by Cletus Emmanuel (cemmanu@uvi.edu), Richard Pinch (R.G.E.Pinch@pmms.cam.ac.uk) %K A004203 nonn %O A004203 0,2 %A A004203 rkg@cpsc.ucalgary.ca (Richard Guy) %I A058029 %S A058029 3,5,11,59,419,839,2521,27733,360337,720703,12252259,232792559, %T A058029 5354228879,26771144401,80313433231,2329089562799,72201776446801, %U A058029 144403552893599,5342931457063157,219060189739591153 %N A058029 Primes closest to LCM[1,...,x] either above or below. Arguments x were selected from A000961 (powers of primes including primes) in order to obtain distinct values of LCM exactly once. %Y A058029 A000961, A003418, A051451. %K A058029 nonn %O A058029 0,1 %A A058029 Labos E. (labos@ana1.sote.hu), Nov 16 2000 %I A046928 %S A046928 3,5,11,71,29,191,137,659,419,809,4721,3119,1667,521,1151,1931,4547, %T A046928 1949,6449,19181,10529,12071,46769,10037,28277,16139,14591,20807, %U A046928 90437,10007,40637,28349,35729,48731,329801,135977,86627,121787 %N A046928 Smaller of twin prime pairs in consecutively larger seas of composite numbers. %e A046928 The pair (3,5) is in a sea _,3-5,6 consisting of a single composite number; the pair (5,7) is in a sea 4,5-7,8,9,10 with 4 composites; (11,13) in a sea 8,9,10,11-13,14,15,16 of 6 composites; etc. %K A046928 nonn %O A046928 1,1 %A A046928 dww %I A062601 %S A062601 3,5,11,157,2557 %N A062601 k^n - (k-1)^n is prime, where k is 35. %C A062601 Terms > 1000 are often only strong pseudo-primes. %Y A062601 Cf. A000043, A057468, A059801, A059802, A062572-A062666. %K A062601 nonn,hard %O A062601 0,1 %A A062601 Mike Oakes (Mikeoakes2@aol.com), May 18 2001, May 19 2001 %I A038198 %S A038198 1,3,5,11,181 %N A038198 n^2+7 is a power of 2. %D A038198 L. J. Mordell, Diophantine Equations, Academic Press, NY, 1969, p. 205. %K A038198 nonn,fini,full %O A038198 0,2 %A A038198 njas %I A037221 %S A037221 3,5,12,10,60,24,135,222,329,139,1749,454,2716,3817 %N A037221 Near-rings (or nearrings) definable on cyclic group of order n. %H A037221 W. E. Clark and J. Pedersen, Catalogue of Algebraic Systems %H A037221 C. Noebauer, Home page %H A037221 C. Noebauer, The Numbers of Small Rings %H A037221 C. Noebauer, Thesis on the enumeration of near-rings %Y A037221 Cf. A027623, A037291. %K A037221 nonn,nice %O A037221 2,1 %A A037221 njas %E A037221 Corrected by Christof Noebauer (christof.noebauer@algebra.uni-linz.ac.at), Sep 29 2000 %I A057587 %S A057587 0,1,3,5,12,15,30,34,60,65,105,111,168,175,252,260,360,369,495, %T A057587 505,660,671,858,870,1092,1105,1365,1379,1680,1695,2040,2056,2448, %U A057587 2465,2907,2925,3420,3439,3990,4010,4620,4641,5313,5335,6072,6095 %N A057587 Nonnegative numbers of form n*(n^2+-1)/2. %K A057587 nonn %O A057587 0,3 %A A057587 njas, Oct 05 2000 %I A032438 %S A032438 0,0,3,5,12,16,27,33,48,56,75,85,108,120,147,161,192,208, %T A032438 243,261,300,320,363,385,432,456,507,533,588,616,675,705, %U A032438 768,800,867,901,972,1008,1083,1121,1200,1240,1323,1365 %N A032438 n^2-floor( (n+1)/2 )^2. %C A032438 The answer to a question from Mike and Laurie Crain (2crains@concentric.net): how many even numbers are there in an n X n multiplication table starting at 1X1? %K A032438 nonn,easy,nice %O A032438 0,3 %A A032438 njas %I A025083 %S A025083 3,5,12,16,34,42,77,91,160,184,312,352,587,653,1076,1184,1938,2114,3445, %T A025083 3731,6064,6528,10592,11344,18387,19605,31756,33728,54610,57802,93565,98731, %U A025083 159792,168152,272120,285648,462235,484125,783380,818800,1324898,1382210 %N A025083 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (Fibonacci numbers), t = (odd natural numbers). %K A025083 nonn %O A025083 1,1 %A A025083 Clark Kimberling (ck6@cedar.evansville.edu) %I A024696 %S A024696 0,0,3,5,12,18,24,30,47,55,82,96,127,149,186,210,261,293,319,349,412,444, %T A024696 517,557,634,680,773,819,918,976,1079,1147,1268,1324,1455,1535,1680,1758, %U A024696 1844,1918,2075,2167,2334,2422,2607,2703,2890,3002,3213,3311,3522,3634 %N A024696 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k = [ (n+1)/2 ], s = A023534, t = (primes). %K A024696 nonn %O A024696 1,3 %A A024696 Clark Kimberling (ck6@cedar.evansville.edu) %I A025088 %S A025088 3,5,12,18,38,52,95,115,206,248,426,494,827,969,1602,1848,3028,3348,5461, %T A025088 6123,9958,10836,17590,18970,30757,33619,54464,59522,96380,103718,167899, %U A025088 177531,287336,307392,497462,524884,849381,894117,1446820,1542058,2495214 %N A025088 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (Fibonacci numbers), t = (primes). %K A025088 nonn %O A025088 1,1 %A A025088 Clark Kimberling (ck6@cedar.evansville.edu) %I A010067 %S A010067 1,3,5,12,19,26,33,40,42,49,56,63,70,77,79,86,93,100,107,114,116, %T A010067 123,130,137,144,151,153,160,167,174,181,188,190,197,204,211,218, %U A010067 229,236,243,250,257,259 %N A010067 Base 6 self or Columbian numbers (not of form n + sum of base 6 digits of n). %K A010067 nonn,base %O A010067 1,2 %A A010067 Leonid Broukhis (leo@mailcom.com) %I A024458 %S A024458 1,1,3,5,12,19,40,65,130,210,404,654,1227,1985,3653,5911,10720,17345, %T A024458 31090,50305,89316,144516,254568,411900,720757,1166209,2029095,3283145, %U A024458 5684340,9197455,15855964,25655489,44061862,71293590,122032508 %N A024458 a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)s(n+1-k), where k = [ (n+1)/2 ], s = (Fibonacci numbers). %K A024458 nonn %O A024458 1,3 %A A024458 Clark Kimberling (ck6@cedar.evansville.edu) %E A024458 More terms from James A. Sellers (sellersj@math.psu.edu), May 03 2000 %I A034763 %S A034763 1,3,5,12,21,60,145,461,1455,4920,16825,58937,208049,743218,2674539, %T A034763 9695792,35357723,129647876,477638761,1767273109,6564120913, %U A034763 24466300714,91482563719,343059732922,1289904147511,4861946817610 %N A034763 Dirichlet convolution of primes (with 1) with Catalan numbers. %K A034763 nonn %O A034763 1,2 %A A034763 Erich Friedman (erich.friedman@stetson.edu) %I A013498 %S A013498 1,0,1,3,5,12,22,50,95,210,406,882,1722,3696,7260,15444,30459, %T A013498 64350,127270,267410,529958,1108536,2200276,4585308,9111830, %U A013498 18929092,37650172,78004500,155266100,320932800,639191160 %N A013498 Permutations in S_n with a certain property. %D A013498 Frank Schmidt; Rodica Simion, Card shuffling and a transformation on S_n, Aequationes Math. 44 (1992), no. 1, 11-34. %p A013498 binomial(n-2,floor(n/2)-1) + floor((n-1)/2)*binomial(n-3,floor(n/2)-2); %K A013498 nonn,easy %O A013498 1,4 %A A013498 njas %I A034758 %S A034758 1,3,5,12,22,70,216,909,4165,21207,116004,678741,4213634,27644897, %T A034758 190899424,1382960388,10480142200,82864878340,682076806220, %U A034758 5832742247551,51724158236078,474869816388803,4506715738447402 %N A034758 Dirichlet convolution of primes (with 1) with Bell numbers. %K A034758 nonn %O A034758 1,2 %A A034758 Erich Friedman (erich.friedman@stetson.edu) %I A036657 %S A036657 0,0,0,1,1,3,5,12,23,52,109,244,532,1196,2671,6055,13726,31380, %T A036657 71901,165635,382610,887215,2062777,4810230,11243898,26346341,61863991, %U A036657 145560102,343121676,810246372,1916417479,4539722153,10769366928 %N A036657 n-node rooted unlabeled trees with out-degree <=2 and exactly 2 edges at the root. %D A036657 S. J. Cyvin et al., Enumeration of constitutional isomers of polyenes, J. Molec. Structure (Theochem), 357 (1995), 255-261. %H A036657 Index entries for sequences related to rooted trees %F A036657 G.f. satisfies A(x) = (W(x)-x)*(1-x)/x-x, where W(x) is g.f. for A001190. %p A036657 N:=40: G036657:=series(G001190*(1/x-1)-1,x,N); A036657:=n->coeff(G036657,x,n); %Y A036657 First differences of A001190. %K A036657 nonn %O A036657 0,6 %A A036657 njas %I A047761 %S A047761 0,1,1,3,5,12,23,55,114,273,588,1428,3156,7752,17427,43263,98516, %T A047761 246675,567281,1430715,3316521,8414640,19633796,50067108,117464424, %U A047761 300830572,709098696,1822766520,4313876888,11124755664,26421284043 %N A047761 A047760(2n+1). %K A047761 nonn %O A047761 0,4 %A A047761 njas %I A026786 %S A026786 1,1,3,5,12,24,53,117,246,580,1178,2916,5768,14834,28731,76221, %T A026786 145108,395048,741392,2063104,3825418,10847078,19907156,57373672, %U A026786 104370554,305110106,550816506,1630489090,2924018194,8751851866 %N A026786 T(n,[ n/2 ]), T given by A026780. %K A026786 nonn %O A026786 0,3 %A A026786 Clark Kimberling, ck6@cedar.evansville.edu %I A027246 %S A027246 1,1,3,5,12,24,53,117,246,580,1178,2916,6150,14834,32656,76221, %T A027246 173719,395048,926664,2063104,4958556,10847078,26619438,59372770, %U A027246 143365880,326086492,774562478,1792293014,4197344582,9861375614 %N A027246 a(n) = greatest number in row n of array T given by A026780. %K A027246 nonn %O A027246 0,3 %A A027246 Clark Kimberling, ck6@cedar.evansville.edu %I A030270 %S A030270 1,1,3,5,12,24,58,128,309,717,1731,4109,9920,23780,57410,138192, %T A030270 333625,804457,1942131,4686341,11313828,27308256,65927962,159150320, %U A030270 384222861,927562581,2239334163,5406150125,13051600952,31509157004 %N A030270 Nonisomorphic and nonantiisomorphic reflexive transitive and cotransitive (complement is transitive) relations. %F A030270 a(n) = (A001333(n) + A001333(floor((n+2)/2))) / 2. %K A030270 nonn,easy %O A030270 0,3 %A A030270 Christian G. Bower (bowerc@usa.net) %I A017921 %S A017921 1,3,5,12,25,56,125,280,625,1398,3125,6988,15625,34939, %T A017921 78125,174693,390625,873465,1953125,4367321,9765625,21836602, %U A017921 48828125,109183007,244140625,545915034,1220703125,2729575168 %N A017921 Powers of sqrt(5) rounded up. %K A017921 nonn %O A017921 0,2 %A A017921 njas %I A005913 %S A005913 1,3,5,12,27,62,143,331,766,1774,4109,9518,22048,51074, %T A005913 118313,274073,634893,1470737,3406980,7892311,18282636, %U A005913 42351953,98108825,227270312,526474502,1219584727,2825183178 %N A005913 a(n) = [ tau*a(n-1) ] + [ tau*a(n-2) ]. %K A005913 nonn %O A005913 0,2 %A A005913 njas %I A056690 %S A056690 0,0,1,1,3,5,12,27,62,149,371,952,2513,6807,18873,53472,154554,455076, %T A056690 1363287,4150560,12829558,40227005,127843589,411512640,1340747909, %U A056690 4418886387,14724633149,49582195801,168638834537,579103599979 %N A056690 floor[product_{k=2 to n}[ln(k)]]. %e A056690 ln(2)*ln(3)*ln(4)*ln(5)= 1.699... So a(5) = 1, since the integer part of 1.699... is 1. %K A056690 easy,huge,nonn %O A056690 2,5 %A A056690 Leroy Quet (qqquet@mindspring.com), Aug 10 2000 %I A046091 %S A046091 1,1,1,3,5,12,30,79,227,709,2318 %N A046091 Connected planar graphs with n edges. %H A046091 E. W. Weisstein, Link to a section of The World of Mathematics. %H A046091 E. W. Weisstein, Link to a section of The World of Mathematics. %K A046091 nonn,nice,hard %O A046091 0,4 %A A046091 Brendan McKay (bdm@cs.anu.edu.au) %I A002905 M2486 N0985 %S A002905 1,1,1,3,5,12,30,79,227,710,2322,8071,29503,112822,450141,1867871, %T A002905 8037472,35787667,164551477,779945969 %N A002905 Connected graphs with n edges. %D A002905 G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817. %D A002905 M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to $p = 18$ Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967. %H A002905 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A002905 G. Royle, Small graphs %H A002905 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A002905 Column sums of A054924 or equivalently row sums of A054923. %Y A002905 Cf. A000664, A046091. %K A002905 nonn,nice %O A002905 1,4 %A A002905 njas %E A002905 More terms from Vladeta Jovovic (vladeta@Eunet.yu), Jan 12 2000 %I A028268 %S A028268 1,3,5,13,7,19,45,9,71,43,105,161,11,53,201,265,615,89,341,1617,15,103, %T A028268 1221,2497,3795,533,1651,8151,17,651,5369,14443,151,785,2835,7553, %U A028268 15379,24453,30745,169,1105,21,1293,5661,229,1501,24225,23,251,8455 %N A028268 Distinct odd elements in 3-Pascal triangle A028262 (by row). %K A028268 nonn,tabf %O A028268 0,2 %A A028268 Mohammad K. Azarian (ma3@cedar.evansville.edu) %E A028268 More terms from James A. Sellers (sellersj@math.psu.edu) %I A018753 %S A018753 1,3,5,13,15,25,39,65,75,195,325,975 %N A018753 Divisors of 975. %K A018753 nonn,fini,full %O A018753 0,2 %A A018753 njas %I A063484 %S A063484 3,5,13,15,35,37,39,55,61,63,73,87,134,155,157,183,193,203,209,219,247, %T A063484 249,259,277,295,305,313,314,327,329,339,341,371,397,399,413,421,457, %U A063484 458,471,489,515,535,539,541,545,579,583,613,635,649,661,673,685,689 %N A063484 Omega(n+1) = 2*Omega(n), where Omega(n) is the number of prime divisors of n (with repetition). %e A063484 Omega(949)=2, Omega(949+1)=2*2, so 949 is a member of the sequence. %o A063484 (PARI.2.0.17) j=[];for(n=1,1000,if(bigomega(n+1)==2*bigomega(n),j=concat(j,n)));j %K A063484 nonn %O A063484 1,1 %A A063484 Jason Earls (jcearls@kskc.net), Jul 28 2001 %I A057920 %S A057920 1,3,5,13,15,35,37,61,73,104,119,157,164,193,194,255,277,313,397,421, %T A057920 455,457,495,527,541,545,584,613,629,661,665,673,733,757,877,975,997, %U A057920 1085,1093,1153,1201,1213,1237,1295,1321,1381,1453,1469,1621,1657,1753 %N A057920 phi(n+1) divides phi(n), where phi(n) is number of positive integers <= n and relatively prime to n. %e A057920 13 is included because phi(14) = 6 divides phi(13) = 12. %K A057920 nonn %O A057920 0,2 %A A057920 Leroy Quet (qqquet@mindspring.com), Nov 11 2000 %I A018329 %S A018329 1,3,5,13,15,39,65,195 %N A018329 Divisors of 195. %K A018329 nonn,fini,full %O A018329 0,2 %A A018329 njas %I A045411 %S A045411 3,5,13,17,19,29,31,41,43,47,59,61,71,73,83,89,97,101,103,113,127,131, %T A045411 139,157,167,173,181,197,199,211,223,227,229,239,241,251,257,269,271, %U A045411 281,283,293,307,311,313,337,349 %N A045411 Primes congruent to {1, 3, 5, 6} mod 7. %K A045411 nonn %O A045411 0,1 %A A045411 njas %I A049282 %S A049282 3,5,13,17,19,31,37,41,53,59,67,71,89,103,107,109,113,131,139,157,163, %T A049282 179,181,193,197,199,211,229,233,239,251,257,269,271,283,293,307,311, %U A049282 337,347,379,383,397,401,409,419,431,433,449,463,467,487,491,499,503 %N A049282 Primes p such that both p-2 and p+2 are square-free. %F A049282 Intersection[ A049231 and A049233 ] %e A049282 37 is here because neither 37+2 nor 37-2 is divisible by squares. %Y A049282 A049231, A049233. %K A049282 nonn %O A049282 1,1 %A A049282 Labos E. (labos@ana1.sote.hu) %I A045412 %S A045412 3,5,13,17,19,31,41,47,59,61,73,83,89,97,101,103,131,139,157,167,173, %T A045412 181,199,223,227,229,241,251,257,269,271,283,293,307,311,313,349,353, %U A045412 367,383,397,409,419,433,439,461 %N A045412 Primes congruent to {3, 5, 6} mod 7. %K A045412 nonn %O A045412 0,1 %A A045412 njas %I A003625 M2487 %S A003625 3,5,13,17,19,31,41,47,59,61,73,83,89,97,101,103,131,139,157,167,173, %T A003625 181,199,223,227,229,241,251,257,269,271,283,293,307,311,313,349,353, %U A003625 367,383,397,409,419,433,439,461,467,479,503,509,521,523,563,577,587,593 %N A003625 Inert rational primes in Q(sqrt -7). %D A003625 H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498. %F A003625 Congruent to 3, 5 or 13 (mod 14). %K A003625 nonn %O A003625 1,1 %A A003625 njas,mb %I A060192 %S A060192 3,5,13,17,29,31,43,47,61,67,79,83,101,103,113,127,139,149,163, %T A060192 167,181,191,199,211,229,233,251,257,271,277,293,307,317,331,349, %U A060192 353,373,379,397,401,421,431,443,449,463,467,491,499,521,523,557 %N A060192 Union_i p(4i+2), p(4i+3), where p(k) = k-th prime. %Y A060192 Cf. A060191, A060193, A060194. %K A060192 nonn %O A060192 0,1 %A A060192 njas, Mar 21 2001 %I A065311 %S A065311 3,5,13,17,29,31,43,67,71,97,107,109,131,157,181,191,223,233,239,269, %T A065311 281,313,359,379,383,401,409,431,503,523,569,571,619,631,659,691,719, %U A065311 751,787,797,857,859,881,883,971,1039,1061,1063,1091,1117,1123,1201 %N A065311 Primes which occur exactly twice in the sequence of a(n)=p[n]-p[n-Pi(n)]=A065308(n). %C A065311 In A065308 each odd primes seems to appear once or twice. %Y A065311 A000040, A000720, A054546, A065308-A065311. %K A065311 nonn %O A065311 1,1 %A A065311 Labos E. (labos@ana1.sote.hu), Oct 29 2001 %I A040158 %S A040158 3,5,13,17,31,37,47,53,59,61,71,73,89,97,101,109,113,127,131,137,139, %T A040158 149,151,157,163,167,179,181,193,211,223,229,239,241,271,281,293,307, %U A040158 311,313,331,337,347,353,373,389 %N A040158 x^4 = 23 has no solution mod p. %K A040158 nonn %O A040158 0,1 %A A040158 njas %I A038185 %S A038185 1,3,5,13,17,59,81,219,257,899,1349,3437,4353,15235,20805,56173,65537, %T A038185 229379,344069,876557,1118225,3913787,5313617,14399195,16842753, %U A038185 58949635,88424453,225271821,285282321 %N A038185 One-dimensional cellular automaton 'sigma' (rule 150). %C A038185 Generation n (starting from the generation 0: 1) cut after the central 1-column and interpreted as a binary number. %H A038185 Index entries for sequences related to cellular automata %p A038185 bit_n := (x,n) -> mod(floor(x/(2^n)),2); %p A038185 sigmacut:=proc(n): if (0 = n) then (1) %p A038185 else sum('((bit_n(sigmagen(n-1),i+1+n-1)+bit_n(sigmagen(n-1),i+n-1)+bit_n(sigmagen(n-1),i-1+n-1)) mod 2)*(2^i)', 'i'=0..(n)) fi: end: %Y A038185 Cf. A006977, A006978, A038183; a(n) = floor(A038184[ n ]/2^n) %K A038185 nonn %O A038185 0,2 %A A038185 Antti Karttunen (karttu@megabaud.fi), 9. Feb 1999 %I A002716 M2488 N0986 %S A002716 3,5,13,17,241,257,65281,65537,4294901761,4294967297, %T A002716 18446744069414584321,18446744073709551617 %N A002716 An infinite coprime sequence. %D A002716 A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422. %K A002716 nonn %O A002716 0,1 %A A002716 njas %E A002716 More terms from J. O. Shallit. %I A046154 %S A046154 3,5,13,17,449,577,193,1153,86017,26214401,114689,7681,147457, %T A046154 754974721 %N A046154 Primes of the form n*2^phi(n)+1 with phi the Euler function. %K A046154 hard,nonn %O A046154 1,1 %A A046154 Felice Russo (felice.russo@katamail.com) %I A045413 %S A045413 3,5,13,19,23,29,43,53,59,73,79,83,89,103,109,113,139,149,163,173,179, %T A045413 193,199,223,229,233,239,263,269,283,293,313,349,353,359,373,379,383, %U A045413 389,409,419,433,439,443,449,463 %N A045413 Primes congruent to {0, 3, 4} mod 5. %K A045413 nonn %O A045413 0,1 %A A045413 njas %I A019420 %S A019420 3,5,13,19,29,37,43,53,59,67,83,101,107,109,131,139,149,173,179,181,197, %T A019420 227,229,251,269,277,283,293,307,317,331,347,373,379,389,397,419,421,443, %U A019420 461,467,491,499,509,523,541,557,563,587,619,643,653,659,661,677,683,691 %N A019420 Primes with primitive root 98. %K A019420 nonn %O A019420 1,1 %A A019420 dww %I A019358 %S A019358 3,5,13,19,29,37,53,59,67,83,107,139,149,163,173,179,197,227,269,293,317, %T A019358 347,349,373,379,389,419,443,467,509,523,547,557,563,587,613,619,653,659, %U A019358 677,709,757,773,787,797,827,829,853,859,877,883,907,947,1019,1109,1117 %N A019358 Primes with primitive root 32. %K A019358 nonn %O A019358 1,1 %A A019358 dww %I A024820 %S A024820 3,5,13,19,33,41,61,85,99,129,163,181,221,265,313,339,393,451,513,545,613, %T A024820 685,761,841,883,969,1059,1153,1251,1301,1405,1513,1625,1741,1861,1923,2049, %U A024820 2179,2313,2451,2593,2665,2813,2965,3121,3281,3445,3613,3699,3873,4051,4233 %N A024820 a(n) = least m such that if r and s in {1/2, 1/4, 1/6,..., 1/2n} satisfy r > s, then r > k/m > s for some integer k. %K A024820 nonn %O A024820 2,1 %A A024820 Clark Kimberling (ck6@cedar.evansville.edu) %I A038941 %S A038941 3,5,13,19,41,47,61,73,83,97,103,107,109,113,127,131,137, %T A038941 149,163,167,179,197,199,229,239,241,257,263,269,271,283, %U A038941 293,317,347,353,367,379,431,439,443,449,461,463,479,487 %N A038941 61 is a square mod p. %K A038941 nonn %O A038941 0,1 %A A038941 njas %I A034484 %S A034484 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,3,5,13, %T A034484 20,44,76,147,253,472,810,1482,2579,4715,8510,16147,30994,62711, %U A034484 130101,281192,620304,1399545,3185945 %N A034484 Multiplicity of highest weight (or singular) vectors associated with character chi_96 of Monster module. %D A034484 K. Harada and M. L. Lung, Modular forms associated with the Monster module, pp. 59-83 of The Monster and Lie Algebras, de Gruyter, 1998. %K A034484 nonn %O A034484 1,31 %A A034484 njas %I A059872 %S A059872 1,3,5,13,21,46,51,52,78,83,84,175,181,205,210,303,309,333,338,390,392, %T A059872 639,698,726,728,737,822,824,846,851,852,903,905,1143,1145,1197,1202, %U A059872 1226,1232,1311,1322,1328,1350,1352,1409,1562,1571,1572,1601,2539,2540 %N A059872 Solutions to the equation given in A059871, encoded as binary vectors and converted to decimal. %C A059872 The rows of this table have lengths given by A059871[n]: 1;3;5;13;21;46,51,52;78,83,84;175,181,205,210; etc... %C A059872 In binary encodings, the least significant bit (bit-0) stands for the factor of 1, the next bit (bit-1) stands for the factor of 2, bit-2 for the factor of 3, bit-3 for the factor of 5, etc, each bit being 0 if the corresponding factor is -1, and 1 if it is +1 (or +2 if the bit is the most significant bit of the code of odd length). %C A059872 E.g. we have 2 = 2*1 -> 1 in binary, 3 = 1*2 + 1*1 -> 11 in binary, 5 = 2*3 - 1*2 + 1*1 -> 101 in binary, 7 = 1*5 + 1*3 - 1*2 + 1*1 -> 1101 in binary, 11 = 2*7 - 1*5 + 1*3 - 1*2 + 1*1 -> 10101 in binary. Function bin_prime_sum given in A059876 maps such encodings back to primes. %p A059872 map(op,primesums_primes_mult(16)); # primesums_primes_mult given in A059871. %Y A059872 Cf. A059873-A059875. %K A059872 nonn,tabf %O A059872 1,2 %A A059872 Antti.Karttunen@iki.fi Feb 05 2001 %I A059873 %S A059873 1,3,5,13,21,46,78,175,303,639,1143,2539,4542,9214,17406,36735,69374, %T A059873 139254,270327,556031,1079294,2162678,4259819,8642558,17022974, %U A059873 34078590,67632893,136249338,270401534 %N A059873 The lexicographically first sequence of binary encodings of solutions satisfying the equation given in A059871. %C A059873 The encoding is explained in A059872. Apply bin_prime_sum (see A059876) to this sequence, and you get A000040, the prime numbers. %p A059873 primesums_primes_search(16); primesums_primes_search := (upto_n) -> primesums_primes_search_aux([],1,upto_n); primesums_primes_search_aux := proc(a,n,upto_n) local i,p,t; if(n > upto_n) then RETURN(a); fi; p := ithprime(n); for i from (2^(n-1)) to ((2^n)-1) do t := bin_prime_sum(i); if(t = p) then print([op(a),i]); RETURN(primesums_primes_search_aux([op(a),i],n+1,upto_n)); fi; od; RETURN([op(a),and no more found]); end; %Y A059873 Cf. A059459, A059874, A059875. %K A059873 nonn %O A059873 1,2 %A A059873 Antti.Karttunen@iki.fi Feb 05 2001 %E A059873 More terms from Naohiro Nomoto (6284968128@geocities.co.jp), Sep 12 2001 %I A059874 %S A059874 1,3,5,13,21,51,83,175,303,639,1143,2539,4571,9711,17903,36735,69499, %T A059874 143339,270327,556031,1080315,2195431,4259819,8646647,17031163, %U A059874 34078647,67632893,136282091,270467055 %N A059874 The lexicographically first sequence of binary encodings of solutions satisfying the equation p_i = (1+mod(i,2))*p_{i-1} +- p_{i-2} +- p_{i-3} +- ... +- 2 + 1, where p_i is the i-th prime number. %C A059874 I.e. like A059873, but the encodings must be "odd", with the least significant bit set. I don't know whether this sequence can be extended infinitely. %Y A059874 Cf. A059873, A059875. %K A059874 nonn %O A059874 1,2 %A A059874 Antti.Karttunen@iki.fi Feb 05 2001 %E A059874 More terms from Naohiro Nomoto (6284968128@geocities.co.jp), Sep 12 2001 %I A059875 %S A059875 1,3,5,13,21,52,84,210,392,905,1601,3652,7173,15364,28932,61952,122900, %T A059875 253969,493572,1017858,2031636,4128801,8159232,16547841,33030657, %U A059875 66584836,132251649 %N A059875 The lexicographically last sequence of binary encodings of solutions satisfying the equation given in A059871. %C A059875 Apply bin_prime_sum (see A059876) to this sequence, and you get A000040, the prime numbers. %p A059875 map(last_term,primesums_primes_mult(16)); last_term:=proc(l) local n: n := nops(l); if(0 = n) then ([]) else (op(n,l)): fi: end: # primesums_primes_mult given in A059871. %Y A059875 Cf. A059873, A059874. %K A059875 nonn %O A059875 1,2 %A A059875 Antti.Karttunen@iki.fi Feb 05 2001 %E A059875 More terms from Naohiro Nomoto (6284968128@geocities.co.jp), Sep 12 2001 %I A014437 %S A014437 1,1,3,5,13,21,55,89,233,377,987,1597,4181,6765,17711,28657, %T A014437 75025,121393,317811,514229,1346269,2178309,5702887,9227465, %U A014437 24157817,39088169,102334155,165580141,433494437,701408733 %N A014437 Odd Fibonacci numbers. %F A014437 Fibonacci(3n+1) union Fibonacci(3n+2). %F A014437 a(n) = Fibonacci(3*floor((n+1)/2)) + (-1)^n) - Antti Karttunen (karttu@walrus.megabaud.fi), Feb 05, 2001 %Y A014437 Cf. A001651, A059878. %K A014437 nonn,easy %O A014437 0,3 %A A014437 Mohammad K. Azarian (ma3@cedar.evansville.edu) %I A045414 %S A045414 3,5,13,23,43,53,73,83,103,113,163,173,193,223,233,263,283,293,313,353, %T A045414 373,383,433,443,463,503,523,563,593,613,643,653,673,683,733,743,773, %U A045414 823,853,863,883,953,983,1013 %N A045414 Primes congruent to {0, 3} mod 5. %K A045414 nonn %O A045414 0,1 %A A045414 njas %I A026733 %S A026733 1,1,3,5,13,23,57,103,249,455,1083,1993,4693,8679,20275,37633, %T A026733 87377,162643,375789,701075,1613413,3015563,6916957,12948083, %U A026733 29617161,55513327,126678893,237705547,541325021,1016736115,2311294377 %N A026733 a(n) = T(n,0) + T(n,1) +... + T(n,[ n/2 ]), T given by A026725. %K A026733 nonn %O A026733 0,3 %A A026733 Clark Kimberling, ck6@cedar.evansville.edu %I A005824 M2489 %S A005824 0,1,1,3,5,13,23,59,105,269,479,1227,2185,5597,9967,25531,45465, %T A005824 116461,207391,531243,946025,2423293,4315343,11053979,19684665,50423309,89792639 %N A005824 a(n) = 5a(n-2) - 2a(n-4). %D A005824 Shallit, Jeffrey; On the worst case of three algorithms for computing the Jacobi symbol. J. Symbolic Comput. 10 (1990), no. 6, 593-610. %K A005824 nonn %O A005824 0,4 %A A005824 njas,jos %I A027305 %S A027305 1,3,5,13,24,58,111,257,500,1126,2210,4882,9632,20980,41531,89497, %T A027305 177564,379438,754014,1600406,3184016,6720748,13382710,28117498, %U A027305 56026984,117254268,233765636,487589572,972504704,2022568168 %N A027305 a(n) = SUM{(k+1)(T(n,k)}, 0<=k<=[ (n+1)/2 ], T given by A008315. %K A027305 nonn %O A027305 0,2 %A A027305 Clark Kimberling, ck6@cedar.evansville.edu %I A026766 %S A026766 1,1,3,5,13,24,59,115,273,552,1278,2655,6031,12795,28632,61775, %T A026766 136572,298764,653948,1447225,3141427,7020833,15132512,34106865, %U A026766 73069892,165903082,353576829,807957495,1714132308,3939206346 %N A026766 a(n) = T(n,0) + T(n,1) + ... + T(n,[ n/2 ]), T given by A026758. %K A026766 nonn %O A026766 0,3 %A A026766 Clark Kimberling, ck6@cedar.evansville.edu %I A026709 %S A026709 1,1,3,5,13,24,61,116,293,564,1421,2752,6925,13457,33839,65891, %T A026709 165621,322905,811431,1583317,3978065,7766507,19511079,38106295, %U A026709 95723669,187002147,469727963,917808733,2305345429,4505028968 %N A026709 T(n,[ n/2 ]), T given by A026703. %K A026709 nonn %O A026709 0,3 %A A026709 Clark Kimberling, ck6@cedar.evansville.edu %I A026720 %S A026720 1,1,3,5,13,25,63,128,319,664,1647,3468,8583,18179,44941,95486, %T A026720 235913,502130,1240173,2642373,6524919,13910933,34346785,73254357, %U A026720 180855499,385819274,952494047,2032271705,5017037457,10705592360 %N A026720 T(n,[ n/2 ]), T given by A026714. %K A026720 nonn %O A026720 0,3 %A A026720 Clark Kimberling, ck6@cedar.evansville.edu %I A026003 %S A026003 1,1,3,5,13,25,63,129,321,681,1683,3653,8989,19825,48439,108545,265729, %T A026003 598417,1462563,3317445,8097453,18474633,45046719,103274625,251595969, %U A026003 579168825,1409933619,3256957317,7923848253,18359266785,44642381823 %N A026003 a(n) = T(n,[ n/2 ]), where T = Delannoy triangle (A008288). %K A026003 nonn %O A026003 0,3 %A A026003 Clark Kimberling (ck6@cedar.evansville.edu) %I A000631 M2490 N0987 %S A000631 1,1,3,5,13,27,66,153,377,914,2281,5690,14397,36564,93650,240916, %T A000631 623338,1619346,4224993,11062046,29062341,76581151,202365823,536113477 %N A000631 Ethylene derivatives with n carbon atoms. %D A000631 A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976, p. 28. %D A000631 H. R. Henze and C. M. Blair, The number of structural isomers of the more important types of aliphatic compounds, J. Amer. Chem. Soc., 56 (1934), 157. %D A000631 H. R. Henze and C. M. Blair, The number of structurally isomeric hydrocarbons of the ethylene series, J. Amer. Chem. Soc., 55 (1933), 680-685. %D A000631 R. C. Read, Some recent results in chemical enumeration, Lect. Notes Math. 303 (1972), 243-259. %K A000631 nonn %O A000631 2,3 %A A000631 njas %I A026569 %S A026569 1,1,3,5,13,27,67,153,375,893,2189,5319,13089,32155,79479,196573, %T A026569 487833,1212135,3018355,7525585,18792303,46980373,117589689,294613155, %U A026569 738844719,1854484305,4658460165,11710592711,29458662005,74151824271 %N A026569 a(n)=T(n,n), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=0. %H A026569 J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5. %K A026569 nonn %O A026569 0,3 %A A026569 Clark Kimberling, ck6@cedar.evansville.edu %I A035082 %S A035082 0,1,0,1,1,3,5,13,27,67,157,390,963,2437,6186,15908,41127,107148, %T A035082 280569,738675,1953054,5185364,13816018,36934431,99030038,266254593, %U A035082 717652816,1938831589,5249221790,14240130827,38702218134,105367669062 %N A035082 Rooted polygonal cacti (Husimi graphs) with n nodes. %D A035082 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, %D A035082 F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134-141 %D A035082 F. Harary and E. M. Palmer, Graphical Enumeration, p. 71 %D A035082 F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953 %H A035082 Index entries for sequences related to cacti %H A035082 Index entries for sequences related to rooted trees %H A035082 N. J. A. Sloane, Transforms %H A035082 C. G. Bower, Transforms (2) %F A035082 Shifts left under transform T where Ta = EULER(BIK(a)-a). %Y A035082 Cf. A003080, A035083-A035088. %K A035082 nonn,eigen %O A035082 0,6 %A A035082 Christian G. Bower (bowerc@usa.net), Nov, 1998 %I A005198 M2491 %S A005198 0,1,1,3,5,13,27,68,160,404,1010,2604,6726,17661,46628,124287, %T A005198 333162,898921,2437254,6640537,18166568,49890419,137478389,380031868 %N A005198 Forests of planted trees. %D A005198 E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121. %K A005198 nonn %O A005198 1,4 %A A005198 njas %I A065047 %S A065047 3,5,13,29,37,43,71,83,101,113,163,193,211,223,257,311,317,347,479,509, %T A065047 547,577,613,643,673,709,787,823,853,877,883,907,1031,1061,1181,1223, %U A065047 1259,1283,1409,1451,1481,1493,1499,1511,1523,1559,1583,1721,1871,1973 %N A065047 Primes which when written in base 2 and prepended with a 1 produce a prime. %e A065047 13 is in the sequence because 13d = 1101b, prepend a one gives 11101b = 29d which is a prime. %t A065047 Do[p = Prime[n]; d = IntegerDigits[p, 2]; If[ PrimeQ[ FromDigits[ Prepend[d, 1], 2]], Print[p]], {n, 1, 350} ] %Y A065047 Cf. A059459. %K A065047 base,easy,nonn %O A065047 1,1 %A A065047 Robert G. Wilson v (rgwv@kspaint.com), Nov 05 2001 %I A051401 %S A051401 3,5,13,31,110,114,197,199,443,659,661,665,1105,1106,1109,1637,2769, %T A051401 2770,2778,2791,2794,2795,2797,2802,2803,6986,6987,7013,7021,9717,9718, %U A051401 9719,9721,9726,9741,9749,9822,9823,9830,9831,9833,9857,9861,23833 %N A051401 Smallest value of x such that M(x) = -n, where M(x) is Mertens's function A002321. %e A051401 M(31) = -4, and that is the first one, so a(4) = 31. %t A051401 s=0;t=0;Do[s=s+MoebiusMu[n];If[sIndex entries for sequences related to rooted trees %F A032009 Shifts left under "AFK" (ordered, size, unlabeled) transform %K A032009 nonn,eigen %O A032009 1,4 %A A032009 Christian G. Bower (bowerc@usa.net) %I A032027 %S A032027 1,1,1,3,5,13,35,95,255,715,2081,6003,17645,52127,155863,468129, %T A032027 1415521,4301055,13134789,40275109,123970669,382919917,1186475687, %U A032027 3686899725,11487023793,35876838669,112304155021,352276801491 %N A032027 Planted planar trees (n+1 nodes) where any 2 subtrees extending from the same node are different. %H A032027 Index entries for sequences related to rooted trees %F A032027 Shifts left under "AGK" (ordered, elements, unlabeled) transform. %K A032027 nonn,eigen %O A032027 1,4 %A A032027 Christian G. Bower (bowerc@usa.net) %I A005383 M2492 %S A005383 3,5,13,37,61,73,157,193,277,313,397,421,457,541,613,661,673,733, %T A005383 757,877,997,1093,1153,1201,1213,1237,1321,1381,1453,1621,1657,1753,1873,1933,1993 %N A005383 n and (n+1)/2 are prime. %C A005383 Appears to be identical to the sequence of numbers n such that sigma(n)/2 is prime. A proof would be most welcome. - Joseph L. Pe (joseph_l_pe@hotmail.com), Dec 10 2001 %D A005383 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870. %Y A005383 Cf. A005382, A057326, A057327, A057328, A057329, A057330, A005603. %K A005383 nonn,easy,more %O A005383 1,1 %A A005383 njas %I A057188 %S A057188 3,5,13,43,79,101,107,227,353,7393 %N A057188 (22^n+1)/23 is a prime (or in some cases, a probable prime). %H A057188 H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7. %K A057188 nonn %O A057188 1,1 %A A057188 njas, Sep 15 2000 %I A034375 %S A034375 1,1,3,5,13,55,377,1597,17711 %N A034375 Fibonacci numbers with all odd digits (probably finite). %K A034375 easy,nonn,base %O A034375 0,3 %A A034375 Erich Friedman (erich.friedman@stetson.edu) %I A051901 %S A051901 3,5,13,73,241,2161,15121,282241,1088641,10886401,199584001,958003201, %T A051901 18681062401,1133317785601,9153720576001,648606486528001, %U A051901 1778437140480001,12804747411456001,851515702861824001 %N A051901 Minimal factorial safe-primes: a p=a[n] here if (p-1)/n! = A051888[n]. %F A051901 a[n] = 1+n!*A051888[n]=1+A000142[n]*A051888[n] %e A051901 a[8]= 282241 = 8!*A051888[8]= 40320*7+1 %Y A051901 A005385, A051888, A000142. %K A051901 nonn %O A051901 1,1 %A A051901 Labos E. (labos@ana1.sote.hu), Dec 16 1999 %I A018928 %S A018928 3,5,13,85,157,12325,12461,106285,276341,339709,10363909,17238541, %T A018928 1936511509,51335823965,133473142309,872709007405,1574530008629, %U A018928 667511933218429,698925273030725,707670964169285,1839944506840141 %N A018928 Define {b(n)} by b(1)=3, b(n) (n >= 2) is smallest number such that b(1)^2+...+b(n)^2 = m^2 for some m and all b(i) are distinct. Sequence gives values of m. %Y A018928 Cf. A018929, A018930. %K A018928 nonn %O A018928 1,1 %A A018928 charles.reed@bbs.ewgateway.org (Charles Reed) %E A018928 More terms from dww. %I A053630 %S A053630 3,5,13,85,3613,6526885,21300113901613,226847426110843688722000885, %T A053630 25729877366557343481074291996721923093306518970391613 %N A053630 Pythagorean spiral: a(n-1), a(n)-1 and a(n) are sides of a right angled triangle. %C A053630 a(3)=85 because 13,84,85 is a Pythagorean triplet and a(2)=13 %F A053630 a(n) = (a(n-1)^2+1)/2 %Y A053630 Cf. A001844. %K A053630 nonn,huge %O A053630 0,1 %A A053630 Henry Bottomley (se16@btinternet.com), Mar 21 2000 %E A053630 Corrected and extended by James A. Sellers (sellersj@math.psu.edu), Mar 22 2000. %I A062698 %S A062698 3,5,13,241,958003201,12804747411456001, %T A062698 20666295932772289859333302675046400000001, %U A062698 3102237506574764560448486032938606422126519440033972224000000000001 %N A062698 Primes of form 2n!+1. %o A062698 (PARI.2.0.17) for(n=1,55,if(isprime(2*n!+1),print(2*n!+1))) %K A062698 huge,nonn %O A062698 1,1 %A A062698 Jason Earls (jcearls@kskc.net), Jul 10 2001 %I A028942 %S A028942 0,0,1,3,5,14,8,69,435,2065,3612,28888,43355,2616119,28076979, %T A028942 332513754,331948240,8280062505,641260644409,18784454671297, %U A028942 318128427505160,10663732503571536,66316334575107447,8938035295591025771 %V A028942 0,0,1,3,5,-14,-8,69,435,2065,3612,-28888,43355,2616119,28076979, %W A028942 -332513754,-331948240,8280062505,641260644409,18784454671297, %X A028942 318128427505160,-10663732503571536,-66316334575107447,8938035295591025771 %N A028942 Numerator of Y-coordinate of n*P where P is generator for rational points on curve y^2+y = x^3-x. %D A028942 B. Mazur, Arithmetic on curves, Bull. Amer Math. Soc. 14 (1986), 207-259; see p 225. %F A028942 P=(0,0), 2P=(1,0), if kP=(a,b) then (k+1)P=(a'=(b^2-a^3)/a^2,b'=1-b*a'/a). %K A028942 sign,done %O A028942 0,4 %A A028942 njas %I A026645 %S A026645 1,1,3,5,14,21,55,85,216,341,848,1365,3340,5461,13191,21845,52208, %T A026645 87381,206968,349525,821514,1398101,3264044,5592405,12979006, %U A026645 22369621,51642594,89478485,205592744,357913941,818848135,1431655765 %N A026645 a(n) = T(n,0) + T(n,1) + ... + T(n,[ n/2 ]), T given by A026637. %K A026645 nonn %O A026645 0,3 %A A026645 Clark Kimberling, ck6@cedar.evansville.edu %I A026667 %S A026667 1,1,3,5,14,21,55,98,255,421,1047,1939,4851,8393,20265,38423, %T A026667 93728,166999,395317,761724,1824934,3319575,7745123,15103234, %U A026667 35691596,65946113,152151529,299476048,700050056,1309537115,2994340063 %N A026667 a(n) = T(n,0) + T(n,1) + ... + T(n,[ n/2 ]), T given by A026659. %K A026667 nonn %O A026667 0,3 %A A026667 Clark Kimberling, ck6@cedar.evansville.edu %I A026777 %S A026777 1,1,3,5,14,26,70,138,362,742,1912,4028,10249,22033,55547,121273, %T A026777 303641,670997,1671233,3729071,9250099,20803231,51437219,116436313, %U A026777 287152067,653567143,1608416195,3677760541,9035150126,20741496354 %N A026777 a(n) = T(n,0) + T(n,1) + ... + T(n,[ n/2 ]), T given by A026769. %K A026777 nonn %O A026777 0,3 %A A026777 Clark Kimberling, ck6@cedar.evansville.edu %I A007136 M2493 %S A007136 0,0,0,1,1,3,5,14,27,65,142,338,773,1832,4296,10231,24296,58128,139132, %T A007136 334350,804441,1940239,4685806,11335797,27455949,66585170,161646826 %N A007136 Symmetry sites in all planted 3-trees with n nodes. %D A007136 Bailey, C. K.; Fundamental orbits. Proceedings of the second West Coast conference on combinatorics, graph theory, and computing (Eugene, Ore., 1983). Congr. Numer. 41 (1984), 149-152. %D A007136 Bailey, C. K.; Palmer, E. M.; Kennedy, J. W.; Points by degree and orbit size in chemical trees. II. Discrete Appl. Math. 5 (1983), no. 2, 157-164. %D A007136 R. W. Robinson, personal communication. %H A007136 Index entries for sequences related to rooted trees %H A007136 Index entries for sequences related to trees %K A007136 nonn %O A007136 1,6 %A A007136 njas %I A052974 %S A052974 1,0,1,3,5,14,34,81,200,487,1187,2899,7072,17256,42109,102748,250717, %T A052974 611779,1492805,3642610,8888370,21688597,52922564,129136875,315108171, %U A052974 768898587,1876197092,4578127192,11171133721,27258794552,66514455833 %N A052974 A simple regular expression. %H A052974 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1046 %F A052974 G.f.: -(-1+2*x)/(1-2*x-x^3+2*x^4-x^2) %F A052974 Recurrence: {a(1)=0,a(0)=1,a(2)=1,a(3)=3,2*a(n)-a(n+1)-a(n+2)-2*a(n+3)+a(n+4)} %F A052974 Sum(1/4999*(-159+1343*_alpha-450*_alpha^2+136*_alpha^3)*_alpha^(-1-n),_alpha=RootOf(1-2*_Z-_Z^3+2*_Z^4-_Z^2)) %p A052974 spec:= [S,{S=Sequence(Prod(Union(Sequence(Union(Z,Z)),Z),Z,Z))},unlabelled ]: seq(combstruct[count ](spec,size=n),n=0..20); %K A052974 easy,nonn %O A052974 0,4 %A A052974 encyclopedia@pommard.inria.fr, Jan 25 2000 %E A052974 More terms from James A. Sellers (sellersj@math.psu.edu), Jun 06 2000 %I A006395 M2494 %S A006395 1,1,3,5,14,42,150,624 %N A006395 Connected planar maps with n nodes. %D A006395 Walsh, T. R. S.; Generating nonisomorphic maps without storing them. SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178. %K A006395 nonn,more %O A006395 1,3 %A A006395 njas %I A058220 %S A058220 1,3,5,15,5,59,159,189,569,105,1557,2549,2439,13797,25353 %N A058220 Ultra-useful primes: smallest k such that 2^(2^n) - k is prime. %Y A058220 Cf. A058221. %K A058220 nonn,hard,nice %O A058220 1,2 %A A058220 Warren Smith (wds@research.nj.nec.com), Nov 30 2000 %I A018358 %S A018358 1,3,5,15,17,51,85,255 %N A018358 Divisors of 255. %K A018358 nonn,fini,full %O A018358 0,2 %A A018358 njas %I A003527 %S A003527 1,3,5,15,17,51,85,255,257,771,1285,3855,4369,13107,21845, %T A003527 65535 %N A003527 Divisors of 2^16 - 1. %K A003527 nonn,fini,full %O A003527 0,2 %A A003527 njas %I A004729 %S A004729 1,3,5,15,17,51,85,255,257,771,1285,3855,4369,13107,21845,65535,65537, %T A004729 196611,327685,983055,1114129,3342387,5570645,16711935,16843009,50529027, %U A004729 84215045,252645135,286331153,858993459,1431655765,4294967295 %N A004729 Divisors of 2^32 - 1 (polygons with an odd number of sides constructible with ruler and compass). %C A004729 The 32 divisors of the product of the 5 known Fermat primes. %C A004729 The only known odd numbers whose totient is a power of 2 - Labos E. (labos@ana1.sote.hu), Dec 06 2000 %D A004729 John H. Conway, Richard K. Guy, The Book of Numbers, Copernicus Press, p. 140. %H A004729 E. W. Weisstein, Link to a section of The World of Mathematics. %t A004729 Divisors[2^32-1] %Y A004729 Essentially same as A045544. %Y A004729 Cf. A000010, A000215, A001317, A003401, A003527, A004169, A004729, A019434, A045544, A047999, A053576, A054432. %K A004729 nonn,fini,full,easy %O A004729 0,2 %A A004729 njas %I A045544 %S A045544 3,5,15,17,51,85,255,257,771,1285,3855,4369,13107,21845,65535,65537, %T A045544 196611,327685,983055,1114129,3342387,5570645,16711935,16843009, %U A045544 50529027,84215045,252645135,286331153,858993459,1431655765,4294967295 %N A045544 Regular n-gons (n odd) which can be constructed by compass and straightedge. %C A045544 If there are no more Fermat primes, then 4294967295 is the last term in the sequence. %H A045544 R. Ballinger and W. Keller, More information %F A045544 Each term is the product of distinct odd Fermat primes. %Y A045544 Cf. A019434. Essentially same as A004729. %Y A045544 Coincides with A001317 for the first 31 terms only. - Robert G. Wilson v (rgwv@kspaint.com), Dec 22 2001 %K A045544 hard,nonn,nice %O A045544 0,1 %A A045544 Ken Takusagawa (kenta@cs.stanford.edu) %I A001317 M2495 N0988 %S A001317 1,3,5,15,17,51,85,255,257,771,1285,3855,4369,13107,21845,65535, %T A001317 65537,196611,327685,983055,1114129,3342387,5570645,16711935,16843009, %U A001317 50529027,84215045,252645135,286331153,858993459,1431655765,4294967295,4294967297,12884901891,21474836485,64424509455,73014444049,219043332147,365072220245,1095216660735,1103806595329,3311419785987 %N A001317 Pascal's triangle mod 2 converted to decimal. %D A001317 H. W. Gould, Exponential Binomial Coefficient Series. Tech. Rep. 4, Math. Dept., West Virginia Univ., Morgantown, WV, Sept. 1961. %D A001317 R. K. Guy, The second strong law of small numbers. Math. Mag. 63 (1990), no. 1, 3-20. %D A001317 D. Hewgill, A relationship between Pascal's triangle and Fermat numbers, Fib. Quart., 15 (1977), 183-184. %H A001317 Index entries for sequences related to cellular automata %p A001317 A001317:=proc(n) local k; add((binomial(n,k) mod 2)*2^k, k=0..n); end; %Y A001317 Cf. A000215 (Fermat numbers). Odd-numbered terms give A038183 (1D Cellular Automata rule 90, "sigma minus") %Y A001317 Not the same as A053576 nor as A045544. %Y A001317 Cf. A047999, A054432. %K A001317 nonn,base,easy,nice %O A001317 0,2 %A A001317 njas %I A053576 %S A053576 1,3,5,15,17,51,85,255,257,771,1285,3855,4369,13107,21845,65535,65537, %T A053576 196611,327685,983055,1114129,3342387,5570645,16711935,16843009, %U A053576 50529027,84215045,252645135,286331153,858993459,1431655765,4294967295,8589934592,17179869184,34359738368,68719476736,137438953472,274877906944,549755813888,1099511627776 %N A053576 Smallest numbers whose Euler totient is divisible by 2^n. %e A053576 1,2,4,8,...,131072 divides Phi of 2,3,5,15,....,196611=3*65537 %Y A053576 Cf. A000010, A003401, A001317, A045544, A058213-A058215. %Y A053576 Not the same as A001317. %Y A053576 More odd terms from Jud McCranie 1/25/00 %K A053576 nonn %O A053576 0,2 %A A053576 Labos E. (labos@ana1.sote.hu), Jan 18 2000 %I A054432 %S A054432 1,3,5,15,17,63,85,219,325,1023,1105,4095,5397,13515,21845,65535,70737, %T A054432 262143,333125,890523,1397077,4194303,4527185,16236015,22365525, %U A054432 57521883,88429845,268435455,272962625,1073741823,1431655765 %N A054432 Numbers formed by interpreting the reduced residue set of n (the rows of triangle A054431) as binary numbers. %F A054432 a(n) = rrs2bincode(n+1) # Starting from n = 1. %F A054432 a(4n-1) = (2^2n + 1)*a(2n-1) [think how the reduced residue set of the numbers of the form 4n are formed] %F A054432 For all p's prime, and e's integer > 1, A054432[p^e] = A019320[p^e]*(((2^(p^(e-1)))-1)* ((2^(p-1))-1))/((2^p)-1) %p A054432 rrs2bincode := proc(n) local i,z; z := 0; for i from 1 to n-1 do z := z*2; if (1 = igcd(n,i)) then z := z + 1; fi; od; RETURN(z); end; %Y A054432 Cf. A054431, A054433, A001317. %K A054432 nonn,huge %O A054432 1,2 %A A054432 Antti.Karttunen@iki.fi (karttu@megabaud.fi) %I A016043 %S A016043 1,3,5,15,17,255,257,65355,65357,4294967295,4294967297, %T A016043 18446744073709551615,18446744073709551617, %U A016043 340282366920938463463374607431768211455 %N A016043 2^(2^n) +- 1 without repeats. %K A016043 nonn %O A016043 1,2 %A A016043 Robert G. Wilson v (rgwv@kspaint.com) %I A018374 %S A018374 1,3,5,15,19,57,95,285 %N A018374 Divisors of 285. %K A018374 nonn,fini,full %O A018374 0,2 %A A018374 njas %I A002962 M2496 %S A002962 1,0,3,5,15,19,58 %N A002962 Simple imperfect squared squares of order n. %C A002962 The order of a squared rectangle is the number of squares into which it is divided. %D A002962 C. J. Bouwkamp, personal communication. %Y A002962 Cf. A006983, A002881, A002839, A014530. %K A002962 nonn,hard,nice %O A002962 13,3 %A A002962 njas %I A063185 %S A063185 1,3,5,15,20,124,141,1085,1221,5267,9814,9899,179888,531293,43914936, %T A063185 59108249,129482155,290253117,297264974,329981693,515700524,1791622856, %U A063185 29237782307,289017844013,2187297805011,20282473409970 %N A063185 Engel Expansion of sum( 1/(2+k)^k, {k,0,Infinity}). %C A063185 Shgz(2)=1.4046684715031192197179531135647252212... %t A063185 ToEngel[ x_,n_Integer ]:=Rest@First@Transpose@NestList[ {Ceiling[ 1/# ],#}&[ Times@@#-1 ]&,{1,Abs[ x ]+1},n ] %Y A063185 Cf. A006784 for definition of Engel expansion. %K A063185 nonn,easy %O A063185 1,2 %A A063185 Olivier Gerard (ogerard@ext.jussieu.fr), Jul 10 2001. %I A059528 %S A059528 1,3,5,15,21,27,65,191,327,371,575,1895,2375,2607 %N A059528 (7*3^n + 5)/2 is prime. %t A059528 Do[ If[ PrimeQ[ (7*3^n + 5)/2 ], Print[n] ], {n, 0, 5300} ] %K A059528 nonn %O A059528 1,2 %A A059528 Robert G. Wilson v (rgwv@kspaint.com), Feb 16 2001 %I A057742 %S A057742 1,1,3,5,15,21,35,60,105,140,210,420,420,1155,1365,2310,4620,5460, %T A057742 9240,13860,16380,27720,32760,60060 %N A057742 Maximal order of element of alternating group A_{2n}. %Y A057742 Bisection of A051593. Cf. A057743. %K A057742 nonn,easy,more %O A057742 0,3 %A A057742 njas, Oct 29 2000 %I A006977 M2497 %S A006977 1,3,5,15,23,59,93,239,375,955,1501,3823,6007,15291,24029,61167,96119,244667, %T A006977 384477,978671,1537911,3914683,6151645,15658735,24606583,62634939 %N A006977 Cellular automaton with 000, 001, 010, 011, ..., 111 -> 0,1,1,0,0,1,1,1. %D A006977 M. Le Brun, personal communication. %H A006977 Index entries for sequences related to cellular automata %H A006977 E. W. Weisstein, Link to a section of The World of Mathematics. %K A006977 nonn %O A006977 1,2 %A A006977 njas %I A003549 %S A003549 1,3,5,15,23,69,89,115,267,345,397,445,683,1191,1335,1985, %T A003549 2047,2049,2113,3415,5955,6141,6339,9131,10235,10245,10565, %U A003549 15709,27393,30705,31695,35333,45655,47127,48599,60787 %N A003549 Divisors of 2^44 - 1. %K A003549 nonn,fini %O A003549 0,2 %A A003549 njas %I A018404 %S A018404 1,3,5,15,23,69,115,345 %N A018404 Divisors of 345. %K A018404 nonn,fini,full %O A018404 0,2 %A A018404 njas %I A053928 %S A053928 3,5,15,23,77,173,235,315,1515,2435,7227,23515,54335,97773,243515, %T A053928 2435077,3090673,72497765,74533585 %N A053928 n^2 contains only digits {2,5,9}. %H A053928 P. De Geest, Squares containing at most three distinct digits, Index entries for related sequences %Y A053928 Cf. A053929. %K A053928 nonn,base,more %O A053928 0,1 %A A053928 Patrick De Geest (pdg@worldofnumbers.com), Mar 2000. %I A018272 %S A018272 1,3,5,15,25,75 %N A018272 Divisors of 75. %K A018272 nonn,fini,full %O A018272 0,2 %A A018272 njas %I A018421 %S A018421 1,3,5,15,25,75,125,375 %N A018421 Divisors of 375. %K A018421 nonn,fini,full %O A018421 0,2 %A A018421 njas %I A053351 %S A053351 3,5,15,27,53,137,153,155,185,239,717,735,789,1343,3485,5109,6195, %T A053351 14927 %N A053351 269*2^n+1 is prime. %H A053351 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A053351 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A053351 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %K A053351 hard,nonn %O A053351 0,1 %A A053351 njas, Dec 29 1999 %I A051044 %S A051044 1,1,3,5,15,27,89,165,585,1113,4097,7917,29927,58499,225585,444793, %T A051044 1741521,3457027,13699699,27342421,109420549,219358315,884987529, %U A051044 1780751883,7233519619,14600965705,59656252987,120742510607 %N A051044 Odd values of the PartitionsQ function A000009. %H A051044 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A051044 Cf. A000009, A051005. %K A051044 nonn %O A051044 0,3 %A A051044 Eric W. Weisstein (eric@weisstein.com) %I A003536 %S A003536 1,3,5,15,29,43,87,113,127,129,145,215,339,381,435,565, %T A003536 635,645,1247,1695,1905,3277,3683,3741,4859,5461,6235,9831, %U A003536 11049,14351,14577,16383,16385,18415,18705,24295,27305 %N A003536 Divisors of 2^28 - 1. %K A003536 nonn,fini %O A003536 0,2 %A A003536 njas %I A048738 %S A048738 1,3,5,15,29,83,179,495,1125,7179,19401,46363,124673,302271,809921, %T A048738 1984959,5304947,13110907,34972559,87014349,231756983,579803757, %U A048738 1542417375 %N A048738 Self-avoiding walks on square lattice rotated by pi/4, with wedge angle pi/2. %D A048738 A. J. Guttmann and G. M. Torrie, Critical behaviour at an edge for the SAW and Ising model, J. Phys. A 17 (1984), 3539-3552. %K A048738 nonn,walk %O A048738 1,2 %A A048738 njas %I A018454 %S A018454 1,3,5,15,29,87,145,435 %N A018454 Divisors of 435. %K A018454 nonn,fini,full %O A018454 0,2 %A A018454 njas %I A018470 %S A018470 1,3,5,15,31,93,155,465 %N A018470 Divisors of 465. %K A018470 nonn,fini,full %O A018470 0,2 %A A018470 njas %I A018516 %S A018516 1,3,5,15,37,111,185,555 %N A018516 Divisors of 555. %K A018516 nonn,fini,full %O A018516 0,2 %A A018516 njas %I A018551 %S A018551 1,3,5,15,41,123,205,615 %N A018551 Divisors of 615. %K A018551 nonn,fini,full %O A018551 0,2 %A A018551 njas %I A018568 %S A018568 1,3,5,15,43,129,215,645 %N A018568 Divisors of 645. %K A018568 nonn,fini,full %O A018568 0,2 %A A018568 njas %I A038375 %S A038375 3,5,15,45,189 %N A038375 Maximal number of spanning paths in tournament on n nodes. %D A038375 K B Reid and L W Beineke "Tournaments", pp. 169-204 in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978. %H A038375 Index entries for sequences related to tournaments %K A038375 nonn %O A038375 3,1 %A A038375 njas %I A018601 %S A018601 1,3,5,15,47,141,235,705 %N A018601 Divisors of 705. %K A018601 nonn,fini,full %O A018601 0,2 %A A018601 njas %I A006394 M2498 %S A006394 1,1,3,5,15,52,213,1002 %N A006394 Connected planar maps with n nodes. %D A006394 Walsh, T. R. S.; Generating nonisomorphic maps without storing them. SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178. %K A006394 nonn,more %O A006394 1,3 %A A006394 njas %I A018650 %S A018650 1,3,5,15,53,159,265,795 %N A018650 Divisors of 795. %K A018650 nonn,fini,full %O A018650 0,2 %A A018650 njas %I A018702 %S A018702 1,3,5,15,59,177,295,885 %N A018702 Divisors of 885. %K A018702 nonn,fini,full %O A018702 0,2 %A A018702 njas %I A018719 %S A018719 1,3,5,15,61,183,305,915 %N A018719 Divisors of 915. %K A018719 nonn,fini,full %O A018719 0,2 %A A018719 njas %I A018771 %S A018771 1,3,5,15,67,201,335,1005 %N A018771 Divisors of 1005. %K A018771 nonn,fini,full %O A018771 0,2 %A A018771 njas %I A006593 M2499 %S A006593 3,5,16,12,15,125,24,40,75,48,80,72,84,60,32768,192,144,524288,384,640, %T A006593 9375,168,120,300,1536,520,576,3072,975,2147483648,336,240,1171875,1500,1040 %N A006593 Least number which is side of n Pythagorean triples. %D A006593 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 114. %H A006593 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A006593 Cf. A046081. %K A006593 nonn %O A006593 1,1 %A A006593 njas %I A039782 %S A039782 3,5,16,17,22,36,40,257,65537 %N A039782 phi(n) is equal to the sum of the prime-power components of n-1. %C A039782 Close to Fermat primes (A019434). Next term if it exists is greater than 10^8 - Jud McCranie (jud.mccranie@mindspring.com), Dec 09 1999. %e A039782 phi(36)=12, 35=5^1*7^1, (5^1)+(7^1)=12. %Y A039782 Cf. A000010, A008475. %K A039782 nonn,more %O A039782 1,1 %A A039782 Olivier Gerard (ogerard@ext.jussieu.fr) %I A019096 %S A019096 0,1,3,5,16,25,51,122,305,616,1413,3192,7055,16930,40856,98582, %T A019096 249604 %N A019096 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite CON = CIT-1 H2[ B2Si54O112 ]. %D A019096 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997. %H A019096 G. Thimm, Cycle sequences of crystal structures %K A019096 nonn %O A019096 3,3 %A A019096 Georg Thimm (mgeorg@ntu.edu.sg) %I A038120 %S A038120 1,3,5,16,50,218,937,4494,21517,106401,526132,2623341,13080136, %T A038120 65354995,326546638 %N A038120 Number of distinct n-digit suffixes of base 6 squares not containing digit 0. %K A038120 nonn,base %O A038120 0,2 %A A038120 dww %I A035089 %S A035089 3,5,17,17,97,193,257,257,7681,12289,12289,12289,40961,65537,65537, %T A035089 65537,786433,786433,5767169,7340033,23068673,104857601,167772161, %U A035089 167772161,167772161,469762049,2013265921,3221225473,3221225473 %N A035089 Smallest prime of form 2^n*k+1,i.e. an arithmetical progression with 2^n differences. %H A035089 Index entries for sequences related to primes in arithmetic progressions %e A035089 a(10)=a(11)=12289 because 2^10x12+1 and 2^11x6+1 are equally the smallest primes in progressions with difference 1024 or 2048 resp. %Y A035089 Analogous case is A034694. Fermat-primes (A000215) are incorporated. See also Fermat numbers:A000051. %K A035089 nonn %O A035089 1,1 %A A035089 Labos E. (labos@ana1.sote.hu) %I A040129 %S A040129 3,5,17,19,23,29,37,41,53,59,61,71,73,79,83,89,97,101,109,127,131,139, %T A040129 149,151,191,193,197,227,233,239,241,251,257,263,277,281,283,293,307, %U A040129 313,317,337,353,359,373,389,409 %N A040129 x^4 = 14 has no solution mod p. %K A040129 nonn %O A040129 0,1 %A A040129 njas %I A045415 %S A045415 3,5,17,19,29,31,43,47,59,61,71,73,89,101,103,113,127,131,157,173,197, %T A045415 199,211,227,229,239,241,257,269,271,281,283,311,313,337,353,367,379, %U A045415 383,397,409,421,439,449,463,467 %N A045415 Primes congruent to {1, 3, 5} mod 7. %K A045415 nonn %O A045415 0,1 %A A045415 njas %I A045416 %S A045416 3,5,17,19,31,47,59,61,73,89,101,103,131,157,173,199,227,229,241,257, %T A045416 269,271,283,311,313,353,367,383,397,409,439,467,479,509,521,523,563, %U A045416 577,593,607,619,647,661,677,691 %N A045416 Primes congruent to {3, 5} mod 7. %K A045416 nonn %O A045416 0,1 %A A045416 njas %I A038891 %S A038891 3,5,17,19,31,59,61,67,71,73,79,101,103,107,127,137,149, %T A038891 151,157,167,179,197,211,223,227,229,233,277,307,313,331, %U A038891 349,353,379,383,389,397,431,439,457,461,487,523,541,547 %N A038891 19 is a square mod p. %K A038891 nonn %O A038891 0,1 %A A038891 njas %I A020592 %S A020592 3,5,17,19,83,113 %N A020592 Smallest nonempty set S containing prime divisors of 4k+7 for each k in S. %K A020592 nonn,fini,full %O A020592 1,1 %A A020592 dww %I A027699 %S A027699 3,5,17,23,29,43,53,71,83,89,101,113,139,149,163,197,257,263,269,277, %T A027699 281,293,311,317,337,347,349,353,359,373,383,389,401,449,461,467,479, %U A027699 503,509,523,547,571,593,599,619,643,673,683,691,739,751,773,797,811 %N A027699 Primes with even number of 1's in binary expansion. %Y A027699 Cf. A027697, A066148, A066149. %K A027699 nonn,easy,base %O A027699 0,1 %A A027699 njas %E A027699 More terms from Erich Friedman (erich.friedman@stetson.edu). %I A024862 %S A024862 3,5,17,23,50,62,110,130,205,235,343,385,532,588,780,852,1095,1185,1485, %T A024862 1595,1958,2090,2522,2678,3185,3367,3955,4165,4840,5080,5848,6120,6987,7293, %U A024862 8265,8607,9690,10070,11270,11690,13013,13475,14927,15433,17020,17572,19300 %N A024862 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (natural numbers), t = (odd natural numbers). %K A024862 nonn %O A024862 2,1 %A A024862 Clark Kimberling (ck6@cedar.evansville.edu) %I A025106 %S A025106 3,5,17,23,50,62,119,141,251,289,496,560,939,1045,1729,1903,3122,3406,5559, %T A025106 6021,9795,10545,17120,18336,29731,31701,51361,54551,88338,93502,151367, %U A025106 159725,258523,272049,440272,462160,747883,783301,1267505,1324815,2143698 %N A025106 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (F(2), F(3), F(4), ...), t = (odd natural numbers). %K A025106 nonn %O A025106 1,1 %A A025106 Clark Kimberling (ck6@cedar.evansville.edu) %I A024867 %S A024867 3,5,17,25,56,76,136,164,265,319,475,553,776,894,1198,1354,1755,1933,2443, %T A024867 2685,3316,3586,4352,4682,5605,6029,7119,7613,8878,9422,10892,11496,13183, %U A024867 13921,15847,16657,18836,19762,22210,23304,26039,27227,30267,31595,34948 %N A024867 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (natural numbers), t = (primes). %K A024867 nonn %O A024867 2,1 %A A024867 Clark Kimberling (ck6@cedar.evansville.edu) %I A025111 %S A025111 3,5,17,25,56,76,147,177,321,389,674,784,1321,1549,2571,2967,4876,5390,8809, %T A025111 9879,16081,17501,28426,30656,49727,54357,88083,96267,155902,167768,271617, %U A025111 287197,464867,497311,804854,849218,1374265,1446647,2340937,2495033,4037272 %N A025111 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (F(2), F(3), F(4), ...), t = (primes). %K A025111 nonn %O A025111 1,1 %A A025111 Clark Kimberling (ck6@cedar.evansville.edu) %I A032619 %S A032619 3,5,17,27,35,39,45,47,63,65,83,93,99,113,123,147,149,173,185,203, %T A032619 209,213,219,227,239,249,255,263,267,285,287,303,309,317,327,329,333, %U A032619 363,365,399,413,419,423,435,447,459,465,467,473,489,497,509,515,519 %N A032619 n concatenated with n + 4 is a prime. %Y A032619 Cf. A032609. %K A032619 nonn %O A032619 1,1 %A A032619 Patrick De Geest (pdg@worldofnumbers.com), may 1998. %I A030077 %S A030077 0,0,1,1,3,5,17,28,105,161,670,1001 %N A030077 Take n equally spaced points on circle, connect them by a path with n-1 line segments; sequence gives number of distinct path lengths under action of dihedral group. %D A030077 Computed by Daniel Gittelson (danielg@cortex.ama.ttuhsc.edu) %e A030077 For n=4, the 3 lengths are: 3 boundary edges (length 3), edge-diagonal-edge (2 + sqrt 2) and diagonal-edge-diagonal (1 + 2sqrt 2). For n=5, the 4 edges of the path may include 0,...,4 diagonals, so a(5)=5. %K A030077 nonn,nice,more %O A030077 0,5 %A A030077 Georgios Vasileiou Dalakouras (gbdalako@engin.umich.edu), Daniel Gittelson (danielg@cortex.ama.ttuhsc.edu) %I A058580 %S A058580 1,3,5,17,29,29,99,169,577 %N A058580 a(n) is the least natural number m such that the fractional part of m*(2^0.5) is less than 2^(-n). %C A058580 Since 2^0.5 is irrational such m must exists because for any irrational number a the the sequence a,2a,3a,4a,5a,... is dense modulo 1. %F A058580 a(n) = min m such that m*(2^0.5)-floor(m*(2^0.5)) < 2^(-n) %e A058580 a(7) = 99 because 99*(2^0.5) = 140.00714267... and 0.00714267... < 2^(-7) = 0.0078125 and 99 is the least natural number that satisfies this inequality. %K A058580 nonn %O A058580 1,2 %A A058580 Avi Peretz (njk@netvision.net.il), Dec 25 2000 %I A038703 %S A038703 3,5,17,29,37,127 %N A038703 Primes p such that p^2 mod q is odd, where q is the previous prime. %F A038703 p(k) is in the sequence if p(k)^2 (mod p(k-1)) is odd. %e A038703 The first prime with a prime lower than itself is 3. This squared is 9, which when divided by the previous prime 2 leaves remainder 1, which is odd. So 3 is in the sequence. 11 is not in the sequence because 11^2, when divided by the previous prime 7, leaves a remainder of 121 (mod 7) = 2, which is even. %t A038703 Prime /@ Select[ Range[ 2, 100 ], OddQ[ Mod[ Prime[ # ]^2, Prime[ # - 1 ] ] ] & ] %Y A038703 Cf. A038702. %K A038703 nonn %O A038703 0,1 %A A038703 Neil Fernandez (primeness@borve.demon.co.uk), May 01 2000 %E A038703 The next term if it exists is > 32452843 = 2000000-th prime. Can someone prove this sequence is finite and full? - Olivier Gerard (ogerard@ext.jussieu.fr), Jun 26 2001 %E A038703 More terms from Olivier Gerard (ogerard@ext.jussieu.fr), Jun 26 2001 %I A023226 %S A023226 3,5,17,29,47,53,59,89,107,137,149,227,347,359,383,389,443,479,503,509, %T A023226 557,593,599,617,659,683,743,773,839,857,887,947,953,1049,1097,1187,1217, %U A023226 1259,1307,1319,1373,1409,1433,1493,1499,1607,1613,1637,1667,1697,1709 %N A023226 n and 7n + 8 both prime. %K A023226 nonn %O A023226 1,1 %A A023226 dww %I A038898 %S A038898 3,5,17,31,37,47,53,59,61,71,89,97,109,113,127,131,137, %T A038898 139,149,151,157,163,167,179,181,211,223,229,239,241,271, %U A038898 281,293,307,311,313,331,337,347,373,389,401,421,433,439 %N A038898 23 is not a square mod p. %K A038898 nonn %O A038898 0,1 %A A038898 njas %I A007802 %S A007802 3,5,17,31,127,257,511,683,2047,2731,3277,3641,8191,43691, %T A007802 52429,61681,65537,85489,131071 %N A007802 n such that game of n X n Button Madness need have no solution. %K A007802 nonn %O A007802 1,1 %A A007802 aartb@win.tue.nl (Aart Blokhuis) %I A056816 %S A056816 3,5,17,31,193,653,1201,46499,532159,3311233,128199521, %T A056816 20512526282340991,6036724301884645488192191, %U A056816 2170304813579195568101904406358277391153 %N A056816 Tribonacci primes. %C A056816 Primes within A000213 %Y A056816 Cf. A000213. %K A056816 nonn %O A056816 0,1 %A A056816 Harvey P. Dale (hpd1@is2.nyu.edu), Sep 01 2000 %I A019414 %S A019414 3,5,17,37,47,53,59,61,71,89,97,113,127,137,139,149,163,167,179,211,223, %T A019414 229,239,241,271,293,307,311,313,347,373,389,401,433,439,443,457,463,491, %U A019414 499,521,547,557,569,587,599,613,617,641,661,677,683,691,701,719,733,739 %N A019414 Primes with primitive root 92. %K A019414 nonn %O A019414 1,1 %A A019414 dww %I A001572 M2500 N0989 %S A001572 1,1,1,1,3,5,17,41,127,365,1119,3413,10685,33561,106827,342129, %T A001572 1104347,3584649,11701369,38374065,126395259,417908329,1386618307, %U A001572 4615388353,15407188529,51569669429,173033992311,581905285089,1961034571967 %N A001572 Related to series-parallel networks. %D A001572 J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570. %F A001572 G.f.: 1 - Sum_{k=1..inf} a(k)*x^k = Product_{n=1..inf} (1-x^n)^A000669(n). %K A001572 nonn,easy %O A001572 0,5 %A A001572 njas %I A005142 M2501 %S A005142 1,1,1,3,5,17,44,182,730,4032,25598,212780,2241730,31193324, %T A005142 575252112,14218209962,472740425319,21208887576786,1286099113807999, %U A005142 105567921675718772 %N A005142 Connected bipartite graphs with n nodes. %D A005142 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998. %D A005142 P. Steinbach, Field Guide to Simple Graphs. Design Lab, Albuquerque NM, 1990. %H A005142 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A005142 Cf. A033995. %K A005142 nonn,nice %O A005142 1,4 %A A005142 njas %E A005142 More terms from R. C. Read (rcread@sympatico.ca). %I A006483 M2502 %S A006483 1,3,5,17,49,161,513,1665,5377,17409,56321,182273,589825,1908737, %T A006483 6176769,19988481,64684033,209321985,677380097,2192048129,7093616641,22955425793 %N A006483 Fibonacci(n)*2^n + 1. %D A006483 D. S. Kluk, personal communication. %K A006483 nonn,easy %O A006483 0,2 %A A006483 njas %I A049540 %S A049540 1,1,1,3,5,17,55,267,1467,10371 %N A049540 Number of n-ominoes for a high-dimensional orthoplex. %C A049540 An orthoplex polyominoes does not extend more than two units along any axis. %e A049540 There are a(5)=5 orthoplex pentominoes: the two 3-D pentominoes that fit in a 2x2x2 box, and the three 4-D pentominoes. %Y A049540 A005519 is a superset. %K A049540 hard,nice,nonn %O A049540 1,4 %A A049540 Robert A. Russell (russell@post.harvard.edu) %I A056826 %S A056826 3,5,17,157 %N A056826 Primes of the form (p^p + 1)/(p + 1) where p is a prime. %D A056826 Richard K. Guy, Unsolved Problems in Theory of Numbers, 1994 A3. %t A056826 Do[ If[ PrimeQ[ (Prime[ n ]^Prime[ n ] + 1)/(Prime[ n ] + 1) ], Print[ Prime[ n ] ] ], {n, 1, 213} ] %K A056826 hard,nonn %O A056826 1,1 %A A056826 Robert G. Wilson v (RGWv@kspaint.com), Aug 29 2000 %I A058910 %S A058910 3,5,17,257,641,1217,14593,167809,671233,1314497,180449537,424050817 %N A058910 Least prime number, not already in sequence, such that the product M of it and all prior numbers in sequence satisfies 2^(M+1) = 1 (mod M). %e A058910 a(3)=17, because n=3*5*17=255 and 2^256 = 1 (mod 255) %K A058910 nonn %O A058910 0,1 %A A058910 Joe K. Crump (joecr@microsoft.com), Jan 09 2001 %I A023394 %S A023394 3,5,17,257,641,65537,114689,274177,319489,974849,2424833,6700417, %T A023394 13631489,26017793,45592577,63766529,167772161,825753601,1214251009, %U A023394 6487031809,70525124609,190274191361,646730219521,2710954639361 %N A023394 Prime factors of Fermat numbers (complete up to a(19) = 1214251009). %H A023394 R. Ballinger and W. Keller, More information %Y A023394 Cf. A000215. %K A023394 nonn %O A023394 1,1 %A A023394 dww %I A056130 %S A056130 3,5,17,257,5189,65537,83269,86293,1053953,1066049,1134929,1311749, %T A056130 1380629,16864513,17060929,17909009,18153809,18171217,21251141, %U A056130 22103317,289423441,290455889,290735441,336662789,336925957,340873541 %N A056130 Palindromic primes in bases 2 and 4. %t A056130 Do[If[PrimeQ[n], t = RealDigits[n, 4][[1]]; If[FromDigits[t] == FromDigits[Reverse[t]], s = RealDigits[n, 2][[1]]; If[FromDigits[s] == FromDigits[Reverse[s]], Print[n]]]], {n, 1, 10^8, 2}] %Y A056130 Cf. A016041 and A029972. %K A056130 nonn,base %O A056130 1,1 %A A056130 Robert G. Wilson v (RGWv@kspaint.com), Jul 29 2000 %I A019434 %S A019434 3,5,17,257,65537 %N A019434 Fermat primes: primes of form 2^(2^n) + 1. %D A019434 R. K. Guy, Unsolved Problems in Number Theory, A3. %H A019434 R. Ballinger and W. Keller, More information %Y A019434 Cf. A000215. %K A019434 nonn,nice %O A019434 1,1 %A A019434 njas, dww %E A019434 It is conjectured that there are only 5 terms - certainly 2^(2^n) + 1 is composite for 5<=n<=21. %I A050922 %S A050922 3,5,17,257,65537,641,6700417,274177,67280421310721,59649589127497217, %T A050922 5704689200685129054721,1238926361552897 %N A050922 Triangle in which n-th row gives prime factors of n-th Fermat number 2^(2^n)+1. %H A050922 R. Ballinger and W. Keller, More information %e A050922 3; 5; 17; 257; 65537; 641,6700417; 274177, 67280421310721;... %Y A050922 Cf. A000215, A019434. %K A050922 nonn,easy,tabf,nice,huge %O A050922 0,1 %A A050922 njas, Dec 30 1999 %E A050922 More terms from Larry Reeves (larryr@acm.org), Apr 13 2000. The next term has 62 digits. %I A000215 M2503 N0990 %S A000215 3,5,17,257,65537,4294967297,18446744073709551617,340282366920938463463374607431768211457, %T A000215 115792089237316195423570985008687907853269984665640564039457584007913129639937, %U A000215 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097 %N A000215 Fermat numbers: 2^(2^n) + 1. %C A000215 It is conjectured that just the first 5 numbers in this sequence are primes. %D A000215 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 7. %D A000215 R. K. Guy, Unsolved Problems in Number Theory, A3. %D A000215 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 14. %D A000215 C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966. pp. 36. %D A000215 C. Pomerance, A tale of two sieves, Notices Amer. Math. Soc., 43 (1996), 1473-1485. %H A000215 R. Ballinger and W. Keller, More information %H A000215 E. W. Weisstein, Link to a section of The World of Mathematics. %F A000215 a(0)=3, a(n) = (a(n-1)-1)^2 + 1 %p A000215 A000215:=n->2^(2^n)+1; %Y A000215 Cf. A019434, A050922, A051179. %K A000215 nonn,easy,nice,huge %O A000215 0,1 %A A000215 njas %I A016045 %S A016045 3,5,17,347,13901,128981,128981,113575727,2426256797,137168442221 %N A016045 Prime gaps that appreciate: prime gaps of 2, 4, 6, ... 2n follow these numbers. %D A016045 Steven Kahan, 2-4-6-8...Prime Gaps That Appreciate, J. Rec. Math., Vol. 25 #1, pp. 44-46, 1993 %e A016045 Consider consecutive primes 347, 349, 353, 359, 367. The gaps are 2, 4, 6, 8, so a(4)=347. %K A016045 nonn %O A016045 1,1 %A A016045 Robert G. Wilson v (rgwv@kspaint.com) %E A016045 Corrected and extended by Jud McCranie (jud.mccranie@mindspring.com), Jun 24 2000 %I A007516 M2504 %S A007516 1,3,5,17,65537 %N A007516 (2^2^...^2) (n times) + 1. %D A007516 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 73. %Y A007516 Cf. A004249. %K A007516 nonn,huge %O A007516 0,2 %A A007516 njas, Robert G. Wilson v (rgwv@kspaint.com) %I A039584 %S A039584 0,3,5,18,21,23,30,33,35,52,62,82,97,108,111,113,126,129,131,138,141, %T A039584 143,152,157,180,183,185,198,201,203,210,213,215,232,242,292,310,312, %U A039584 315,317,322,340,350,362,372,375,377,380,392,412,422,442,457,472,490 %N A039584 Representation in base 6 has same number of 1's, 2's and 4's. %K A039584 nonn,base,easy %O A039584 0,2 %A A039584 Olivier Gerard (ogerard@ext.jussieu.fr) %I A011964 %S A011964 0,3,5,18,38,100,198,445,829,1605,2851,5014,8361,13843,21907,34362, %T A011964 52327,78571,115434,167695 %N A011964 Ferrites M_{10}Y_n that repeat after 6n+50 layers. %D A011964 T. J. McLarnan, The numbers of polytypes ..., Zeits. Krist. 155, 269-291 (1981). %K A011964 nonn,easy %O A011964 1,2 %A A011964 njas %I A022489 %S A022489 0,1,1,3,5,18,61,357,2330,18715,167783,1686327,18615873,224310263, %T A022489 2926679435,41112186396,618573033369,9924788849845,169150125371328, %U A022489 3051766556302719 %N A022489 An upper bound for linearized chord diagrams. %H A022489 A. Stoimenow, Enumeration of chord diagrams and an upper bound for Vassiliev invariants, J. Knot Theory Ramifications, 7 (1998), no. 1, 93-114. %K A022489 nonn %O A022489 1,4 %A A022489 stoimeno@informatik.hu-berlin.de (Alexander Stoimenow) %I A053484 %S A053484 1,3,5,19,7,109,331,155,2327,20947,34913,164591,691283,14977801, %T A053484 314533829,4718007451,1572669151,16041225341,103122162907,4571749222213, %U A053484 68576238333199,110777000384399,55582845806909,364345554264288511 %N A053484 Numerators in expansion of exp(2x)/(1-x). %Y A053484 Cf. A053485, A010842. %K A053484 nonn,frac %O A053484 0,2 %A A053484 njas, Jan 15 2000 %I A025046 %S A025046 3,5,19,17,67,71,131,73,277,311,827,241,1607,2543,3691,1559,6803,5711, %T A025046 14969,1009,43103,10559,52057,2689,90313,162263,127403,18191,209327, %U A025046 31391,607153,8089 %N A025046 Least modulus >= 3 having maximum run of n consecutive residues. %K A025046 nonn %O A025046 2,1 %A A025046 dww %I A058778 %S A058778 3,5,19,25,29,47,167,407,3909,4433,4845,4921,30349,78873 %N A058778 n^2*2^n - n*2^((n + 1)/2) + 1 is prime. %D A058778 Posting to NMBRTHRY@LISTSERV.NODAK.EDU by Mile Oakes (Mikeoakes2@aol.com) Dec 30 2000 entitled "Gaussian analogues of the Cullen and Woodall primes." %t A058778 Do[ If[ PrimeQ[ n^2*2^n - n*2^((n + 1)/2) + 1], Print[n] ], {n, 1, 100000, 2} ] %K A058778 nonn %O A058778 1,1 %A A058778 Robert G. Wilson v (rgwv@kspaint.com), Jan 02 2001 %I A062594 %S A062594 3,5,19,31,257,773 %N A062594 k^n - (k-1)^n is prime, where k is 28. %C A062594 Terms > 1000 are often only strong pseudo-primes. %Y A062594 Cf. A000043, A057468, A059801, A059802, A062572-A062666. %K A062594 nonn,hard %O A062594 0,1 %A A062594 Mike Oakes (Mikeoakes2@aol.com), May 18 2001, May 19 2001 %I A034511 %S A034511 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,5,19, %T A034511 42,104,221,496,1011,2134,4334,8951,18305,38126,79788,170261,367939, %U A034511 808612,1798946,4048562,9179756,20926386,47817157 %N A034511 Multiplicity of highest weight (or singular) vectors associated with character chi_123 of Monster module. %D A034511 K. Harada and M. L. Lung, Modular forms associated with the Monster module, pp. 59-83 of The Monster and Lie Algebras, de Gruyter, 1998. %K A034511 nonn %O A034511 1,31 %A A034511 njas %I A055452 %S A055452 1,3,5,19,75,305,1270,5390,23236,101480,448085,1997115,8973255, %T A055452 40602093,184853055,846206025,3892585325,17984308775,83417287855, %U A055452 388297304825,1813341109825,8493372326675,39889629750600 %N A055452 T(n,n-2), array T as in A055450. %K A055452 nonn %O A055452 2,2 %A A055452 Clark Kimberling, ck6@cedar.evansville.edu, May 18 2000 %I A062577 %S A062577 3,5,19,311,317,1129,4253,7699 %N A062577 k^n - (k-1)^n is prime, where k is 11. %C A062577 Terms > 1000 are often only strong pseudo-primes. %Y A062577 Cf. A000043, A057468, A059801, A059802, A062572-A062666. %K A062577 nonn,hard %O A062577 0,1 %A A062577 Mike Oakes (Mikeoakes2@aol.com), May 18 2001, May 19 2001 %I A007363 M2505 %S A007363 0,1,3,5,20,168,11748 %N A007363 Maximal self-dual antichains on n points. %H A007363 D. E. Loeb (loeb@delanet.com), On Games, Voting Schemes, and Distributive Lattices. LaBRI Report 625-93, University of Bordeaux I, 1993. %K A007363 nonn %O A007363 1,3 %A A007363 njas %I A065926 %S A065926 1,3,5,21,25,35,55,185,265,563,569,733,3350,3469,6010 %N A065926 Index values for new maxima in A065925. %o A065926 (PARI) sopf(n) = local(fac, i); fac=factor(n);sum(i=1,matsize(fac)[1],fac[i,1]) A065926(m)= {local(a,n,k);a=0;for(k=1,m,n=1;while(sopf(n)!=sopf(n+k),n++); if(n>a,a=n;print1(k,",")))} --Klaus Brockhaus. %Y A065926 Cf. A065925, A065927. %K A065926 more,nonn %O A065926 1,2 %A A065926 Jason Earls (jcearls@kskc.net), Nov 28 2001 %I A032414 %S A032414 3,5,21,27,59,75,111,287,414,786,966,1071,2433,2817,3165,4958,5895, %T A032414 12450,39399,50019,57386,72599,75866,82026 %N A032414 129*2^n+1 is prime. %H A032414 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A032414 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A032414 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %H A032414 Y. Gallot, Proth.exe: Windows Program for Finding Large Primes %H A032414 R. Ballinger and W. Keller, More information %K A032414 nonn %O A032414 1,1 %A A032414 Jim Buddenhagen (jbuddenh@texas.net) %I A062225 %S A062225 3,5,21,27,95,2075,2165,3047,3503,16791 %N A062225 Smoothly undulating palindromic primes of the form (78*10^a(n)-87)/99. %C A062225 Prime versus probable prime status and proofs are given in the author's table. %H A062225 P. De Geest, SUPP Reference Table %e A062225 a(n)=21 -> (78*10^21-87)/99 = 787878787878787878787. %Y A062225 Cf. A062209-A062232, A059758, A032758. %K A062225 nonn,base %O A062225 0,1 %A A062225 Patrick De Geest (pdg@worldofnumbers.com) and Hans Rosenthal (Hans.Rosenthal@t-online.de), Jun 15 2001. %I A001775 M2506 N0991 %S A001775 1,3,5,21,41,49,89,133,141,165,189,293,305,395,651,665,771,801,923,953, %T A001775 3689,5315,6989,15641 %N A001775 19*2^n-1 is prime. %D A001775 H. Riesel, Prime numbers and computer methods for factorization,'' Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384. %D A001775 H. C. Williams and C. R. Zarnke, Math. Comp., 22 (1968), 420-422. %H A001775 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A001775 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A001775 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %K A001775 hard,nonn %O A001775 1,2 %A A001775 njas %I A056803 %S A056803 3,5,21,69,313 %N A056803 Palindromic primes of the form 12 repeated n times 1. %e A056803 12121212121 is prime so 5 appears in the sequence. %t A056803 Do[m = n; If[PrimeQ[120(10^(2n) - 1)/99 + 1], Print[n]], {n, 1, 600}] %K A056803 nonn %O A056803 1,1 %A A056803 Robert G. Wilson v (RGWv@kspaint.com), Aug 22 2000 %I A025093 %S A025093 3,5,22,30,66,82,161,191,342,680,768,1291,1437,2382,2622,4306,4698,7673, %T A025093 8311,13526,14562,23648,25328,41075,43797,70966,75374,122066,129202,209169, %U A025093 220719,357254,375946,608424,638672,1033531,1082477,1751630,1830830 %N A025093 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (Lucas numbers), t = (odd natural numbers). %K A025093 nonn %O A025093 1,1 %A A025093 Clark Kimberling (ck6@cedar.evansville.edu) %I A025112 %S A025112 3,5,22,30,73,91,172,204,335,385,578,650,917,1015,1368,1496,1947,2109,2670, %T A025112 2870,3553,3795,4612,4900,5863,6201,7322,7714,9005,9455,10928,11440,13107, %U A025112 13685,15558,16206,18297,19019,21340,22140,24703,25585,28402,29370,32453 %N A025112 a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)s(n-k+1), where k = [ n/2 ], s = (odd natural numbers). %K A025112 nonn %O A025112 1,1 %A A025112 Clark Kimberling (ck6@cedar.evansville.edu) %I A025098 %S A025098 3,5,22,32,74,100,199,239,436,530,922,1074,1815,2129,3540,4086,6724,7432, %T A025098 12157,13635,22204,24166,39262,42342,68697,75095,121702,133012,215424, %U A025098 231818,375335,396863,642398,687230,1112246,1173552,1899149,1999177 %N A025098 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (Lucas numbers), t = (primes). %K A025098 nonn %O A025098 1,1 %A A025098 Clark Kimberling (ck6@cedar.evansville.edu) %I A025117 %S A025117 3,5,22,32,81,111,212,256,429,519,794,926,1329,1537,2092,2368,3109,3427, %T A025117 4376,4818,6001,6495,7938,8542,10287,11075,13148,14074,16491,17505,20314, %U A025117 21444,24679,26067,29768,31298,35493,37249,41972,44054,49343,51607,57494 %N A025117 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (odd natural numbers), t = (primes). %K A025117 nonn %O A025117 1,1 %A A025117 Clark Kimberling (ck6@cedar.evansville.edu) %I A064187 %S A064187 3,5,23,5,13,7,7,79,37,23,67,89,13,89,131,31,71,47,43,73,277,353,41,67, %T A064187 127,223,79,13,193,5,23,43,5,67,3,19,5,59,59,653,19,19,97,409,5,383,29, %U A064187 137,379,349,653,1187,47,41,37,17,619,89,283,283,43,479,191 %N A064187 First of n^2 odd consecutive primes whose sum (=S) is divisible by n and S/n = n mod 2. %C A064187 A necessary condition for the existence of a magic square consisting of n^2 consecutive odd primes. %e A064187 a(5)=13 since 13+17+ ... +113 = 1565 = 5*313 and 313 = 5 mod 2. %o A064187 (PARI) for(n=1,50,k=2;m=n^2;aflag=0;while(k+m<=500000&&aflag==0,s=0;for(x=k,k+m-1,s=s+prime(x));if(s%n==0&&(s/n)%2==n%2,print(prime(k));aflag=1);k++)) %Y A064187 Position of 3's gives A064013. %K A064187 nonn %O A064187 1,1 %A A064187 H. K. Gottlob Maier (1korrago@freenet.de), Sep 20, 2001 %I A036952 %S A036952 3,5,23,47,89,101,149,157,163,173,179,185,199,229,247,253,295,313,329, %T A036952 331,355,367,379,383,405,425,443,453,457,471,523,533,539,565,583,587, %U A036952 595,631,643,647,653,659,671,675,689,703,709,755,781,785,815,841,855 %N A036952 Primes with digits (0,1) taken as base 2 and converted to base 10. %C A036952 The sequence is unexpectedly(?) rich in primes. %e A036952 a(n)=313 -> is 100111001{2} -> 100111001{10} is prime. %Y A036952 Cf. A020449, A036953-A036964. %K A036952 nonn,base %O A036952 0,1 %A A036952 Patrick De Geest (pdg@worldofnumbers.com), Jan 1999. %I A065720 %S A065720 3,5,23,47,89,101,149,157,163,173,179,199,229,313,331,367,379,383,443, %T A065720 457,523,587,631,643,647,653,659,709,883,947,997,1009,1091,1097,1163, %U A065720 1259,1277,1283,1289,1321,1483,1601,1669,1693,1709,1753,1877 %N A065720 Primes p such that the decimal expansion of its base 2 conversion is also prime. %C A065720 In general rebase notation (Marc Le Brun): p(n) = (2) [p] (10). %e A065720 E.g. 1009{10} = 1111110001{2} is prime but also 1111110001{10}. %t A065720 Select[ Range[1900], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 2]]] & ] %Y A065720 Cf. A065721 up to A065727, A065361. %K A065720 nonn,base %O A065720 0,1 %A A065720 Patrick De Geest (pdg@worldofnumbers.com), Nov 15 2001. %I A023247 %S A023247 3,5,23,59,73,79,109,179,269,373,383,389,409,439,509,599,683,709,929,983, %T A023247 1019,1129,1193,1409,1423,1453,1663,1699,1879,2039,2053,2069,2579,2753, %U A023247 2963,3049,3169,3203,3259,3719,3769,3833,4799,4973,4993,5303,5443,5483 %N A023247 Remains prime through 2 iterations of function f(x) = 3x + 4. %K A023247 nonn %O A023247 1,1 %A A023247 dww %I A027753 %S A027753 3,5,23,59,113,383,509,653,1193,1409,3083,4973,6323,8933,10103, %T A027753 12659,17033,19463,23873,24809,25763,29759,30803,35159,36293,47309, %U A027753 48623,52673,54059,62753,67343,68909,75353,83813,87323,92723,94559 %N A027753 Primes of form n^2 + n + 3. %H A027753 P. De Geest, World!Of Numbers %K A027753 nonn %O A027753 0,1 %A A027753 Patrick De Geest (pdg@worldofnumbers.com) %I A018978 %S A018978 3,5,23,137,986,6175,36845,230163,1516136 %N A018978 Cycle class sequence c(2n) (the number of true cycles of length 2n in which a certain node is included) for zeolite APC = AlPO4-C [ Al16P16O64 ] (1,2). %D A018978 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997. %H A018978 G. Thimm, Cycle sequences of crystal structures %K A018978 nonn %O A018978 2,1 %A A018978 Georg Thimm (mgeorg@ntu.edu.sg) %I A062218 %S A062218 3,5,23,2177,3147,4157 %N A062218 Smoothly undulating palindromic primes of the form (35*10^a(n)-53)/99. %C A062218 Prime versus probable prime status and proofs are given in the author's table. %H A062218 P. De Geest, SUPP Reference Table %e A062218 a(n)=23 -> (35*10^23-53)/99 = 35353535353535353535353. %Y A062218 Cf. A062209-A062232, A059758, A032758. %K A062218 nonn,base %O A062218 0,1 %A A062218 Patrick De Geest (pdg@worldofnumbers.com) and Hans Rosenthal (Hans.Rosenthal@t-online.de), Jun 15 2001. %I A005761 M2507 %S A005761 1,3,5,24,13,22,13,5,51,76,5,102,12,64,0,127,169,337,135,466, %T A005761 806,234,906,373,368,121,607,942,2422,1292,2847,5294,1871, %U A005761 5062,3241,1240,1089,2500,4164,13362,7521,13590,26201,10290 %V A005761 1,-3,-5,24,-13,-22,13,-5,51,-76,5,102,-12,-64,0,-127,-169,337,135,466, %W A005761 -806,-234,906,-373,-368,-121,-607,-942,2422,1292,2847,-5294,-1871, %X A005761 5062,-3241,-1240,-1089,-2500,-4164,13362,7521,13590,-26201,-10290 %N A005761 Coefficients of modular function G_3(tau). %D A005761 A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14-32. %K A005761 sign,done,easy,nice %O A005761 -3,2 %A A005761 njas %E A005761 More terms from Kok Seng Chua (chuaks@ihpc.nus.edu.sg), Jun 14 2000 %I A009002 %S A009002 1,1,1,3,5,25,61,427,1385,12465,50521,555731,2702765,35135945,199360981, %T A009002 2990414715,19391512145,329655706465,2404879675441,45692713833379,370371188237525, %U A009002 7777794952988025,69348874393137901,1595024111042171723,15514534163557086905 %N A009002 Expansion of (1+x)/cos(x). %t A009002 (1+x)/Cos[ x ] %K A009002 nonn %O A009002 0,4 %A A009002 rhh@research.bell-labs.com %E A009002 Extended and formatted 03/97 by Olivier Gerard %I A054388 %S A054388 1,3,5,28,72,704,1664,2560,17408,311296,688128,3014656,6553600, %T A054388 56623104,121634816,520093696,369098752,7516192768,79456894976, %U A054388 111669149696,704374636544,5909874999296,824633720832,51677046505472 %N A054388 Denominators of coefficients of 1/2^(2n+1) in Newton's series for pi. %H A054388 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A054388 Cf. A054387. %K A054388 nonn %O A054388 0,2 %A A054388 Eric W. Weisstein (eric@weisstein.com) %I A016552 %S A016552 3,5,29,1,17,2,7,5,1,8,9,20,1,2,1,1,1,2,4,1,4,1,4,3,1,1, %T A016552 1,3,1,1,1,1,5,1,22,2,2,31,2,41,1,1,1,1,2,1,7,6,3,2,3,2, %U A016552 1,3,1,8,1,4,1,1,1,4,20,1,1,2,1,4,1,3,4,5,1,2,1,157,1,22 %N A016552 Continued fraction for ln(49/2). %K A016552 nonn,cofr %O A016552 1,1 %A A016552 njas %I A048637 %S A048637 3,5,29,71,83,113,173,263,311,419,431,491,503,509,683,701,761,773,839, %T A048637 911,953,1031,1091,1103,1151,1193,1259,1283,1373,1451,1523,1583,1601, %U A048637 1733,1823,1889,1931,2099,2153,2213,2273,2339,2351,2441,2531,2543,2609 %N A048637 Primes p such that p^3 + 2 is also prime. %e A048637 3^3 + 2 = 29 is prime. %t A048637 Prime[ Select[ Range[ 500], PrimeQ[Prime[ # ]^3 + 2] &] ] %Y A048637 Cf. A048636. %K A048637 easy,nonn %O A048637 3,1 %A A048637 Enoch Haga (EnochHaga@msn.com) %E A048637 More terms from Robert G. Wilson v (rgwv@kspaint.com), Dec 04 2000 %I A035410 %S A035410 3,5,31,39,46,58,63,196,205,210,254,257,266,273,287,296,299,354,361, %T A035410 916,925,930,984,1000,1003,1012,1037,1044 %N A035410 Related to Rogers-Ramanujan Identities. %D A035410 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109. %D A035410 Contact author at address below [ Note: this form of description is not accepted! - NJAS ] %Y A035410 Cf. A003106, A035406, A035407, A035408, A035409. %K A035410 nonn,more,part %O A035410 1,1 %A A035410 Olivier Gerard (ogerard@ext.jussieu.fr) %I A059940 %S A059940 3,5,31,41,107,11,17,727,499,443,863,439,457,3373,23,1637,53,6857,31, %T A059940 47,5323,811,6911,919,29,19681,439,739,13499,29789,43,7187,43,461, %U A059940 23327,50651,59,2579,2909,22973,2179,15901,14197,293,1187,34607,11059 %N A059940 Least prime p such that x = n is a solution mod p of x^3 = 2, or 0 if no such prime exists. %C A059940 Solutions mod p are represented by integers from 0 to p-1. The following equivalences hold for n > 1: There is a prime p such that n is a solution mod p of x^3 = 2 iff n^3-2 has a prime factor > n; n is a solution mod p of x^3 = 2 iff p is a prime factor of n^3-2 and p > n. %C A059940 n^3-2 has at most two prime factors > n, consequently these factors are the only primes p such that n is a solution mod p of x^3 = 2. For n such that n^3-2 has no prime factor > n (the zeros in the sequence; they occur beyond the last entry shown in the database) see A060591. For n such that n^3-2 has two prime factors > n, cf. A060914. %F A059940 If n^3-2 has prime factors > n, then a(n) = the least of these prime factors, else a(n) = 0. %e A059940 a(2) = 3, since 2 is a solution mod 3 of x^3 = 2, and 2 is not a solution mod p of x^3 = 2 for prime p = 2. Although 2^3 = 2 mod 2, prime 2 is excluded because 0 < 2 and 2 = 0 mod 2. a(5) = 41, since 5 is a solution mod 41 of x^3 = 2, and 5 is not a solution mod p of x^3 = 2 for primes p < 41. Although 5^3 = 2 mod 3, prime 3 is excluded because 3 < 5 and 5 = 2 mod 3. %Y A059940 Cf. A040028, A060121, A060122, A060123, A060124, A060591, A060914. %K A059940 nonn %O A059940 2,1 %A A059940 Klaus Brockhaus (klaus-brockhaus@t-online.de), Mar 02 2001 %I A047105 %S A047105 1,3,5,31,120,491,2016,8039,31884,126281,498558,1965077,7740246, %T A047105 30473582,119948507,472129697,1858535320,7317392228,28816569762, %U A047105 113512547183,447270667394,1762899256258,6950527615283,27412071431285 %N A047105 T(n,n-3), array T as in A047100. %K A047105 nonn %O A047105 3,2 %A A047105 Clark Kimberling, ck6@cedar.evansville.edu %I A052468 %S A052468 1,3,5,35,63,77,429,6435,12155,46189,88179,676039,1300075,5014575, %T A052468 646323,300540195,583401555,756261275,4418157975,6892326441, %U A052468 22427411435,263012370465,514589420475,2687300306925,15801325804719 %N A052468 Numerators in the Taylor series for arcosh(x)-ln(2x). %D A052468 Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.2.6 %H A052468 E. W. Weisstein, Link to a section of The World of Mathematics. %e A052468 a(6) = 77 = 1*3*5*7*9*11 / gcd( 1*3*5*7*9*11, 2*4*6*8*10*12*12 ) %Y A052468 a(n) / A052469(n) = A001147(n) / ( A000165(n) *2*n ) %K A052468 nonn,easy,frac %O A052468 1,2 %A A052468 Eric W. Weisstein (eric@weisstein.com) %E A052468 Updated May 22 2001 by Frank.Ellermann@t-online.de %I A055786 %S A055786 1,1,3,5,35,63,231,143,6435,12155,46189,88179,676039,1300075,5014575, %T A055786 9694845,100180065,116680311,2268783825,1472719325,34461632205, %U A055786 67282234305,17534158031,514589420475,8061900920775,5267108601573 %N A055786 Numerators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x). %C A055786 arcsin(x) is usually written as x + x^3/(2*3) + 1*3*x^5/(2*4*5) + 1*3*5*x^7/(2*4*6*7) + ..., = x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ... when reduced to lowest terms. %C A055786 arccos(x) = Pi/2 - (x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ...). %C A055786 arccsc(x) = 1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+... %C A055786 arcsec(x) = Pi/2 -(1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+...) %C A055786 arcsinh(x) = x-1/6*x^3+3/40*x^5-5/112*x^7+35/1152*x^9-63/2816*x^11+... %D A055786 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88. %D A055786 H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, Chap. 3. %e A055786 a(4) = 35 = 3*5*7*9 / gcd( 3*5*7*9, (2*4*6*8) * (2*4+1)) %Y A055786 Cf. A002595. %Y A055786 a(n) / A002595(n) = A001147(n) / ( A000165(n) * (2*n+1)) %K A055786 nonn,frac,nice,easy %O A055786 0,3 %A A055786 njas, Jul 13 2000 %I A001790 M2508 N0992 %S A001790 1,1,3,5,35,63,231,429,6435,12155,46189,88179,676039,1300075,5014575, %T A001790 9694845,300540195,583401555,2268783825,4418157975,34461632205, %U A001790 67282234305,263012370465,514589420475,8061900920775,15801325804719 %N A001790 Numerators in expansion of 1/sqrt(1-x). %D A001790 W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables). %D A001790 H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18. %Y A001790 Cf. A005187. Bisections give A061548 and A063079. %K A001790 nonn,easy,nice %O A001790 1,3 %A A001790 njas %I A057908 %S A057908 1,3,5,35,95,1323 %N A057908 4^n - n is prime. %t A057908 Do[ If[ PrimeQ[ 4^n - n ], Print[ n ] ], {n, 0, 3000} ] %K A057908 nonn %O A057908 1,2 %A A057908 Robert G. Wilson v (rgwv@kspaint.com), Nov 16 2000 %I A062633 %S A062633 3,5,43,2399 %N A062633 k^n - (k-1)^n is prime, where k is 67. %C A062633 Terms > 1000 are often only strong pseudo-primes. %Y A062633 Cf. A000043, A057468, A059801, A059802, A062572-A062666. %K A062633 nonn,hard %O A062633 0,1 %A A062633 Mike Oakes (Mikeoakes2@aol.com), May 18 2001, May 19 2001 %I A056260 %S A056260 3,5,53,95,453,573 %N A056260 Palindromic primes of the form 76(n times)7. %K A056260 hard,nonn,base %O A056260 1,1 %A A056260 Robert G. Wilson v (RGWv@kspaint.com), Aug 18 2000 %I A020462 %S A020462 3,5,53,353,3533,5333,33353,33533,35353,35533,53353,55333,333533,353333, %T A020462 533353,535333,3335533,3353333,3353533,3355553,3533533,3553553,3555353, %U A020462 5333353,5333533,5353553,5533553,33335333,33555553,35535553,35553533 %N A020462 Primes that contain digits 3 and 5 only. %K A020462 nonn,base %O A020462 1,1 %A A020462 dww %I A049190 %S A049190 1,3,5,59,245,2491,235253,127756731,330567489269,258479716298484155, %T A049190 36823182192123209878050549,25576412117054296344209353299113896379, %U A049190 10994511204169842163496446583221775727830456269734123253 %N A049190 Start with 1. Convert to base 2, describe it in base 2, convert to base 2. %e A049190 1->One 1->11->3; 3->11->Two 1s->21->5; 5->21->One 2, One 1->1211->59; etc. %Y A049190 Cf. A005150, A001388. %K A049190 nonn,base %O A049190 1,2 %A A049190 Olivier Gerard (ogerard@ext.jussieu.fr). %I A062214 %S A062214 3,5,77,163,1479,3657,4573,8315 %N A062214 Smoothly undulating palindromic primes of the form (18*10^a(n)-81)/99. %C A062214 Prime versus probable prime status and proofs are given in the author's table. %H A062214 P. De Geest, SUPP Reference Table %e A062214 a(n)=5 -> (18*10^5-81)/99 = 18181. %Y A062214 Cf. A062209-A062232, A059758, A032758. %K A062214 nonn,base %O A062214 0,1 %A A062214 Patrick De Geest (pdg@worldofnumbers.com) and Hans Rosenthal (Hans.Rosenthal@t-online.de), Jun 15 2001. %I A057663 %S A057663 3,5,89,317,701 %N A057663 Primes p such that p+2^p is also a prime. %C A057663 Different from A056206, where e.g. at n=89, 89 is not minimal, A056206(89)=29 and not 89. %Y A057663 Cf. A056206, A056208, A057664, A057665. %K A057663 more,nonn %O A057663 0,1 %A A057663 Labos E. (labos@ana1.sote.hu), Oct 16 2000 %I A056244 %S A056244 1,3,5,93,159,359,1469 %N A056244 Palindromic primes of the form 13(n times)1. %t A056244 Do[If[PrimeQ[(1*10^n + 3*(10^n - 1)/9)*10 + 1], Print[n]], {n, 1, 2500}] %K A056244 hard,nonn,base %O A056244 1,2 %A A056244 Robert G. Wilson v (RGWv@kspaint.com), Aug 18 2000 %I A003112 M2509 %S A003112 1,3,5,105,81,6765,175747,30375,25219857,142901109,4548104883, %T A003112 31152650265,5198937484375 %V A003112 1,-3,-5,-105,81,6765,175747,30375,25219857,142901109,4548104883, %W A003112 -31152650265,-5198937484375 %N A003112 Permanent of Schur's matrix of order 2n+1. %D A003112 Graham, R. L.; Lehmer, D. H. On the permanent of Schur's matrix. J. Austral. Math. Soc. Ser. A 21 (1976), no. 4, 487-497. %D A003112 Lehmer, D. H. Some properties of circulants. J. Number Theory 5 (1973), 43-54. %D A003112 I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 121. %H A003112 E. W. Weisstein, Link to a section of The World of Mathematics. %K A003112 hard,sign,done %O A003112 0,2 %A A003112 njas %I A054266 %S A054266 3,5,109,193,281,509,661,827,857,1439,2111,3433,3889,3967,4549,6661, %T A054266 7001,8467,10099,17203,18583,21011,21611,23831,24847,25117,26261,26497, %U A054266 26861,28181,29587,30497,31307,47569,47869,49789,53939,54139,66361 %N A054266 Sum of composite numbers between prime p and nextprime(p) is palindromic. %H A054266 P. De Geest, World!Of Numbers %e A054266 a(4)=193 since between 193 and next prime 197 we get the palindromic sum 194 + 195 + 196 = 585. %Y A054266 Cf. A046933, A054264, A054265, A054267, A054268. %K A054266 nonn,base %O A054266 0,1 %A A054266 Patrick De Geest (pdg@worldofnumbers.com), Apr 2000. %I A054268 %S A054268 3,5,109,111111109,259259257 %N A054268 Sum of composite numbers between prime p and nextprime(p) is a repdigit. %H A054268 E. W. Weisstein, Repdigit %e A054268 a(5) is ok since between 259259257 and nextprime 259259261 we get the sum 259259258 + 259259259 + 259259260 which yield repdigit 777777777. %Y A054268 Cf. A010785, A028987, A028988, A046933, A054264, A054265, A054266, A054267. %K A054268 nonn,base,hard %O A054268 0,1 %A A054268 Patrick De Geest (pdg@worldofnumbers.com), Apr 2000. %I A038535 %S A038535 1,1,3,5,175,441,4851,14157,2760615,8690825,112285459,370263621, %T A038535 19870814327,67607800225,931331941875,3241035157725,2913690606794775, %U A038535 10313859829588425,147068001273760875,527570807893408125 %V A038535 1,-1,-3,-5,-175,-441,-4851,-14157,-2760615,-8690825,-112285459,-370263621, %W A038535 -19870814327,-67607800225,-931331941875,-3241035157725,-2913690606794775, %X A038535 -10313859829588425,-147068001273760875,-527570807893408125,-30451387031607516975 %N A038535 Numerators of coefficients of EllipticE/Pi. %F A038535 a(n) = 2^(-2 w[n])Binomial[2n,n]^2 (-1)^(2n)/(1-2n) with w[n]=A000120 = number of 1's in binary expansion of n %Y A038535 Cf. A038533, A038534. %K A038535 frac,nonn %O A038535 0,3 %A A038535 Wouter Meeussen (wouter.meeussen@vandemoortele.com), revised Jan 03 2001 %I A012817 %S A012817 1,3,5,217,2505,765963,6838195,9335128657,286658869905, %T A012817 407628336475923,19130907180121195,42164783730598660297, %U A012817 3124552441785986300505,9154657902519091515622683 %V A012817 1,3,5,-217,2505,765963,-6838195,-9335128657,286658869905, %W A012817 407628336475923,-19130907180121195,-42164783730598660297, %X A012817 3124552441785986300505,9154657902519091515622683 %N A012817 arcsinh(sec(x)*sinh(x))=x+3/3!*x^3+5/5!*x^5-217/7!*x^7+2505/9!*x^9... %K A012817 sign,done %O A012817 0,2 %A A012817 Patrick Demichel (dml@hpfrcu03.france.hp.com) %I A058846 %S A058846 3,5,229,257,1031,7213,11863,133853,1002073,31924583,97137589, %T A058846 1837875227 %N A058846 Sum of odd primes up to n is palindromic. %C A058846 Sequence is 3 + 5 + 7 + 11 + 13 + 17 + ... + n. %H A058846 P. De Geest, Palindromic Sums %Y A058846 Cf. A058845, A058847. %K A058846 nonn,base,more %O A058846 1,1 %A A058846 Patrick De Geest (pdg@worldofnumbers.com), Dec 2000. %I A002427 M2510 N0993 %S A002427 1,1,1,1,3,5,691,35,3617,43867,1222277,854513,1181820455,76977927, %T A002427 23749461029,8615841276005,84802531453387,90219075042845,26315271553053477373, %U A002427 38089920879940267,261082718496449122051,1520097643918070802691 %V A002427 1,1,-1,1,-3,5,-691,35,-3617,43867,-1222277,854513,-1181820455,76977927, %W A002427 -23749461029,8615841276005,-84802531453387,90219075042845,-26315271553053477373, %X A002427 38089920879940267,-261082718496449122051,1520097643918070802691 %N A002427 Numerator of (2n+1) B_{2n}, where B_n are the Bernoulli numbers. %D A002427 L. Euler, Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 93. %D A002427 A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 73. %D A002427 M. Kaneko, "A recurrence formula for the Bernoulli numbers", Proc. Japan Acad., 71 A (1995), 192-193. %H A002427 Index entries for sequences related to Bernoulli numbers. %e A002427 (n+1)*B_n gives the sequence 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ... %Y A002427 Denominators are in A006955. Cf. A050925/A050932, A000367/A002445. %K A002427 sign,done,easy,huge,nice,frac %O A002427 0,5 %A A002427 njas %I A012783 %S A012783 1,3,5,889,82167,7869301,906141171,128450716913,22562588189167, %T A012783 4955964959446253,1381444138161821675,495014642393801914473, %U A012783 227710892670360844819175,131704509800404485504904293 %V A012783 1,3,5,-889,-82167,-7869301,-906141171,-128450716913,-22562588189167, %W A012783 -4955964959446253,-1381444138161821675,-495014642393801914473, %X A012783 -227710892670360844819175,-131704509800404485504904293 %N A012783 sin(sec(x)*arcsin(x))=x+3/3!*x^3+5/5!*x^5-889/7!*x^7-82167/9!*x^9... %K A012783 sign,done %O A012783 0,2 %A A012783 Patrick Demichel (dml@hpfrcu03.france.hp.com) %I A062655 %S A062655 3,5,997,4253 %N A062655 k^n - (k-1)^n is prime, where k is 89. %C A062655 Terms > 1000 are often only strong pseudo-primes. %Y A062655 Cf. A000043, A057468, A059801, A059802, A062572-A062666. %K A062655 nonn,hard %O A062655 0,1 %A A062655 Mike Oakes (Mikeoakes2@aol.com), May 18 2001, May 19 2001 %I A038023 %S A038023 0,0,0,0,0,0,3,6,0,4,96,214,128,64,38,54248,83925,54223,28254, %T A038023 10223,1894 %N A038023 Triangle: T(n,k), k<=n: commutative groupoids with a nontrivial symmetry with n elements and k idempotents. %H A038023 Index entries for sequences related to groupoids %F A038023 Difference of A038021 and A038022. %Y A038023 Cf. A001329, A001425, A030259, A030262, A030265. %K A038023 nonn,tabl,more %O A038023 0,7 %A A038023 Christian G. Bower (bowerc@usa.net), May 1998. %I A008954 %S A008954 0,1,3,6,0,5,1,8,6,5,5,6,8,1,5,0,6,3,1,0,0,1,3,6,0,5,1,8,6,5,5,6,8,1, %T A008954 5,0,6,3,1,0,0,1,3,6,0,5,1,8,6,5,5,6,8,1,5,0,6,3,1,0,0,1,3,6,0,5,1, %U A008954 8,6,5,5,6,8,1,5,0,6,3,1,0,0,1,3,6,0,5,1,8,6,5,5,6,8,1,5,0,6,3,1,0 %N A008954 Final digit of triangular numbers n(n+1)/2. %H A008954 Index entries for sequences related to final digits of numbers %F A008954 a(1) = 1, a(n+1) = (a(n) + n + 1) mod 10. %t A008954 Table[ Mod[ n*(n + 1)/2, 10 ], {n, 1, 80} ] %Y A008954 Cf. A000217, A061501. %K A008954 nonn,base,easy %O A008954 0,3 %A A008954 njas %I A021739 %S A021739 0,0,1,3,6,0,5,4,4,2,1,7,6,8,7,0,7,4,8,2,9,9,3,1,9,7,2,7,8,9,1,1,5, %T A021739 6,4,6,2,5,8,5,0,3,4,0,1,3,6,0,5,4,4,2,1,7,6,8,7,0,7,4,8,2,9,9,3,1, %U A021739 9,7,2,7,8,9,1,1,5,6,4,6,2,5,8,5,0,3,4,0,1,3,6,0,5,4,4,2,1,7,6,8,7 %N A021739 Decimal expansion of 1/735. %K A021739 nonn,cons %O A021739 0,4 %A A021739 njas %I A010470 %S A010470 3,6,0,5,5,5,1,2,7,5,4,6,3,9,8,9,2,9,3,1,1,9,2,2,1,2,6,7,4,7,0,4,9, %T A010470 5,9,4,6,2,5,1,2,9,6,5,7,3,8,4,5,2,4,6,2,1,2,7,1,0,4,5,3,0,5,6,2,2, %U A010470 7,1,6,6,9,4,8,2,9,3,0,1,0,4,4,5,2,0,4,6,1,9,0,8,2,0,1,8,4,9,0,7,1 %N A010470 Decimal expansion of square root of 13. %K A010470 nonn,cons %O A010470 1,1 %A A010470 njas %I A011368 %S A011368 1,3,6,0,7,9,0,0,0,0,1,7,4,3,7,6,9,6,4,2,5,5,6,8,0,3,7,5,3,2,4,8,1, %T A011368 3,0,9,8,8,7,0,1,2,6,6,9,8,7,3,9,3,2,7,8,6,3,6,6,4,7,9,9,7,1,4,9,6, %U A011368 6,6,5,3,2,7,6,0,8,8,7,4,6,8,2,5,7,2,1,7,6,2,8,4,1,2,2,9,5,4,6,4,1 %N A011368 Decimal expansion of 9th root of 16. %K A011368 nonn,cons %O A011368 1,2 %A A011368 njas %I A020811 %S A020811 1,3,6,0,8,2,7,6,3,4,8,7,9,5,4,3,3,8,7,8,8,7,3,8,0,0,4,1,5,0,3,2,7, %T A020811 2,9,9,5,5,3,6,6,3,7,4,8,9,2,5,3,7,0,5,6,2,6,9,0,7,0,5,1,4,2,6,2,5, %U A020811 0,5,3,3,5,4,3,0,3,1,8,4,1,6,8,1,4,1,1,0,3,3,0,1,8,3,9,1,4,6,6,7,9 %N A020811 Decimal expansion of 1/sqrt(54). %K A020811 nonn,cons %O A020811 0,2 %A A020811 njas %I A010618 %S A010618 3,6,0,8,8,2,6,0,8,0,1,3,8,6,9,4,6,8,9,2,5,2,5,1,7,2,9,5,9,5,8,8,9, %T A010618 2,6,1,4,9,0,5,5,5,1,6,9,0,1,6,2,3,3,7,8,7,6,8,9,7,9,0,6,0,5,7,8,6, %U A010618 9,4,7,7,9,8,5,9,4,2,1,2,3,7,0,4,9,3,9,1,7,3,7,0,6,6,0,1,5,8,6,9,7 %N A010618 Decimal expansion of cube root of 47. %K A010618 nonn,cons %O A010618 1,1 %A A010618 njas %I A009014 %S A009014 1,0,1,3,6,0,90,630,2520,0,113400,1247400,7484400,0,681080400, %T A009014 10216206000,81729648000,0,12504636144000,237588086736000, %U A009014 2375880867360000,0,548828480360160000,12623055048283680000 %V A009014 1,0,-1,3,-6,0,90,-630,2520,0,-113400,1247400,-7484400,0,681080400, %W A009014 -10216206000,81729648000,0,-12504636144000,237588086736000, %X A009014 -2375880867360000,0,548828480360160000,-12623055048283680000 %N A009014 Has exponential generating function (1+x)/(1+x+x^2/2). %F A009014 E.g.f.: (1+x)/(1+x+x^2/2) = 1/cosh(ln(1+x)). %F A009014 a(n) = -n*a(n-1)-n*(n-1)/2*a(n-2), a(0)=1, a(1)=0. %t A009014 1/Cosh[ Log[ 1+x ] ] %o A009014 (PARI) a(n)=-(2^-n)*n!*real((-1+I)^(n+1)) %K A009014 sign,done,easy %O A009014 0,4 %A A009014 rhh@research.bell-labs.com %E A009014 Extended with signs 03/97 by Olivier Gerard. Better description from Michael Somos (somos@grail.cba.csuohio.edu) %I A009036 %S A009036 1,0,1,3,6,0,114,1134,9320,75456,640144,5888080,59989416,683640048, %T A009036 8694795576,122022500040,1861267622656,30418406611456,526523001326720, %U A009036 9570437327493504,181530307816180736,3575813555711685120 %V A009036 1,0,-1,3,-6,0,114,-1134,9320,-75456,640144,-5888080,59989416,-683640048, %W A009036 8694795576,-122022500040,1861267622656,-30418406611456,526523001326720, %X A009036 -9570437327493504,181530307816180736,-3575813555711685120 %N A009036 Expansion of cos(sin(ln(1+x))). %t A009036 Cos[ Sin[ Log[ 1+x ] ] ] %K A009036 sign,done,easy %O A009036 0,4 %A A009036 rhh@research.bell-labs.com %E A009036 Extended with signs 03/97 by Olivier Gerard. %I A021281 %S A021281 0,0,3,6,1,0,1,0,8,3,0,3,2,4,9,0,9,7,4,7,2,9,2,4,1,8,7,7,2,5,6,3,1, %T A021281 7,6,8,9,5,3,0,6,8,5,9,2,0,5,7,7,6,1,7,3,2,8,5,1,9,8,5,5,5,9,5,6,6, %U A021281 7,8,7,0,0,3,6,1,0,1,0,8,3,0,3,2,4,9,0,9,7,4,7,2,9,2,4,1,8,7,7,2,5 %N A021281 Decimal expansion of 1/277. %K A021281 nonn,cons %O A021281 0,3 %A A021281 njas %I A034004 %S A034004 0,1,3,6,1,0,1,5,2,1,2,8,3,6,4,5,5,5,6,6,7,8,9,1,1,0,5,1,2,0,1,3,6, %T A034004 1,5,3,1,7,1,1,9,0,2,1,0,2,3,1,2,5,3,2,7,6,3,0,0,3,2,5,3,5,1,3,7,8, %U A034004 4,0,6,4,3,5,4,6,5,4,9,6,5,2,8,5,6,1,5,9,5,6,3,0,6,6,6,7,0,3,7,4,1 %N A034004 Successive digits of triangular numbers. %Y A034004 Cf. A000217. %K A034004 nonn %O A034004 0,3 %A A034004 njas %I A016660 %S A016660 3,6,1,0,9,1,7,9,1,2,6,4,4,2,2,4,4,4,4,3,6,8,0,9,5,6,7,1,0,3,1,4,4, %T A016660 7,1,6,3,9,0,0,0,7,7,5,8,7,1,6,7,6,3,6,1,6,3,6,4,4,9,1,2,6,8,1,1,9, %U A016660 2,9,8,9,7,4,6,9,9,0,3,6,1,0,6,5,3,9,9,0,2,1,5,3,3,6,7,2,1,6,8,6,6 %N A016660 Decimal expansion of ln(37). %D A016660 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2. %K A016660 nonn,cons %O A016660 1,1 %A A016660 njas %I A008953 %S A008953 1,3,6,1,1,2,2,3,4,5,6,7,9,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5, %T A008953 6,6,7,7,7,8,8,9,9,9,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2, %U A008953 2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4 %N A008953 Leading digits of triangular numbers n(n+1)/2. %K A008953 nonn,base %O A008953 1,2 %A A008953 njas %I A032660 %S A032660 3,6,1,2,1,4,5,6,59,2,1,4,5,6,1,6,3,2,3,10,7,2,7,4,3,4,1,14,1,2,5,10, %T A032660 1,2,1,2,3,6,7,2,1,14,21,10,11,6,3,4,5,6,1,12,9,34,3,22,11,2,1,2,15,18, %U A032660 1,12,3,2,9,6,1,12,1,2,5,4,5,8,13,8,5,4,7,2,3,2,9,4,5,2,1,8,3,4,1,6,1 %N A032660 Smallest n concatenated with n + d (d = 0,1,2,3,...) is a lucky number. %C A032660 First terms of sequences A032640 to A032649, continued with displacements d > 9. %e A032660 77th term = 13: 13 + 76 (=77-1) = 89: '13' and '89' = 1389 equals a lucky number. %Y A032660 Cf. A000959, A032661. %K A032660 nonn %O A032660 0,1 %A A032660 Patrick De Geest (pdg@worldofnumbers.com), may 1998. %I A055263 %S A055263 0,1,3,6,1,6,3,1,9,9,10,3,6,10,6,3,10,9,9,10,3,6,10,6,3,10,9,9,10,12,6, %T A055263 10,6,12,10,9,9,10,12,6,10,6,12,10,9,9,10,12,6,10,6,12,10,9,9,10,12,15, %U A055263 10,15,12,10,9,9,10,12,15,10,15,12,10,9,9,10,12,15,10,15,12,10,9,9,10 %N A055263 Sum of digits of (n + a(n-1)). %C A055263 If n=0 or 8 mod 9, then a(n)=0 mod 9; if n=1, 4 or 7 mod 9, then a(n)=1 mod 9; if n=2 or 6 mod 9, then a(n)=3 mod 9; if n=3 or 5 mod 9, then a(n)=6 mod 9. %F A055263 a(n) = A007953(A055262(n)) =A007953(n+a(n-1)) %e A055263 a(13)=10 because a(12)=6, 13+6=19 and 1+9=10 %Y A055263 Cf. A055262, A055264. %K A055263 base,easy,nonn %O A055263 0,3 %A A055263 Henry Bottomley (se16@btinternet.com), May 08 2000 %I A004157 %S A004157 1,3,6,1,6,3,10,9,9,10,12,15,10,6,3,10,9,9,10,3,6,10,15,3,10,9,18,10,12, %T A004157 15,19,15,12,19,9,18,10,12,15,10,15,12,19,18,9,10,12,15,10,15,12,19,9,18, %U A004157 10,21,15,10,15,12,19,18,9,10,12,6,19,15,12,19,18,18,10,21,15,19,6,12,10 %N A004157 Sum of digits of triangular numbers. %K A004157 nonn,base %O A004157 1,2 %A A004157 njas %I A065233 %S A065233 1,1,3,6,1,6,12,19,3,12,22,33,45,12,26,41,57,74,17,36,56,77,99,11,35, %T A065233 60,86,113,141,16,46,77,109,142,176,7,43,80,118,157,197,238,19,62,106, %U A065233 151,197,244,292,16,66,117,169,222,276,331,387,48,106,165,225,286,348 %N A065233 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the nonagonal numbers. The first elements of the rows form a(n). %Y A065233 Cf. A064766, A064865, A065221-A065232, A065234. %K A065233 easy,nonn %O A065233 0,3 %A A065233 Floor van Lamoen (f.v.lamoen@wxs.nl), Oct 22 2001 %I A022836 %S A022836 1,3,6,1,8,19,6,23,4,27,56,25,62,21,64,17,70,11,72,5,76,3,82,165,76, %T A022836 173,72,175,68,177,64,191,60,197,58,207,56,213,50,217,44,223,42,233, %U A022836 40,237,38,249,26,253,24,257,18,259,8,265,2,271,542,265,546,263,556 %N A022836 a(n) = c(1)p(i-1)+...+c(n)p(n-1), where c(i) = 1 if a(i-1) >= p(i-1) and c(i) = -1 if a(i-1) < p(i-1) (p(0)=1, p(i)=primes). %K A022836 nonn %O A022836 0,2 %A A022836 Clark Kimberling (ck6@cedar.evansville.edu) %I A026250 %S A026250 3,6,1,10,13,2,17,20,23,4,27,30,5,34,37,40,7,44,47,8,51,54,9, %T A026250 58,61,64,11,68,71,12,75,78,81,14,85,88,15,92,95,16,99,102,105, %U A026250 18,109,112,19,116,119,122,21,126,129,22,133,136,139,24,143,146 %N A026250 Beginning with the natural numbers, swap [ k*sqrt(2) ] and [ k*(2 + sqrt(2)) ], for all k >= 1. %K A026250 nonn %O A026250 1,1 %A A026250 Clark Kimberling, ck6@cedar.evansville.edu %I A004158 %S A004158 1,3,6,1,51,12,82,63,54,55,66,87,19,501,21,631,351,171, %T A004158 91,12,132,352,672,3,523,153,873,604,534,564,694,825,165, %U A004158 595,36,666,307,147,87,28,168,309,649,99,5301,1801 %N A004158 Triangular numbers written backwards. %K A004158 nonn,base %O A004158 1,2 %A A004158 njas %I A058178 %S A058178 1,0,1,0,1,0,1,1,0,1,3,6,2,1,1,16,34,14,11,4,2,858,1453,854,500,174,38, %T A058178 6 %N A058178 Triangle: Self-converse quasigroups of order n with k idempotents. %H A058178 Index entries for sequences related to quasigroups %e A058178 1; 0,1; 0,1,0; 1,1,0,1; 3,6,2,1,1; ... %Y A058178 Row sums give A057993. %K A058178 nonn,tabl,more %O A058178 0,11 %A A058178 Christian G. Bower (bowerc@usa.net), Nov 20 2000 %I A058078 %S A058078 1,1,3,6,2,1,1,35,15,3,2,1,3,5,14,6,6,7,1,1,5,4,4,15,3,1,2,2,55,5,4,3, %T A058078 1,1,3,84,1,1,28,10,3,3,1,1,1,221,3,6,2,7,3,15,231,21,7,1,5,70,3,1, %U A058078 1292,35,1,3,15,24,7,1,6,7,1,3,42,5,1,231,35,1,143,2,5,1,1,7,14,1,45,3 %N A058078 Least common divisor of 2 consecutive special binomial coefficients formed from consecutive primes: a(n)=GCD[C[p(n+2),p(n+1)],C[p(n+1),p(n)]. %e A058078 n=8, a(8)=GCD[C[p(10),p(9)],C[p(9),p(8)]]=GCD[C[29,23],C[23,19]]= =GCD[8855,475020]=GCD[5.7.11.23, 4.9.13.29.5.7]=35 %Y A058078 Cf. A058077. %K A058078 nonn %O A058078 0,3 %A A058078 Labos E. (labos@ana1.sote.hu), Nov 13 2000 %I A016551 %S A016551 3,6,2,1,2,2,2,2,1,1,1,2,141,2,6,19,3,4,2,3,1,4,16,1,4, %T A016551 3,2,6,2,1,2,12,1,1,9,18,2,1,1,12,2,1,1,2,1,13,1,3,1,6, %U A016551 1,3,6,1,39,2,4,2,10,2,13,2,3,2,1,2,1,4,4,3,2,3,2,1,1,4 %N A016551 Continued fraction for ln(47/2). %K A016551 nonn,cofr %O A016551 1,1 %A A016551 njas %I A058099 %S A058099 1,0,3,6,2,2,5,16,12,2,17,10,48,56,10,24,35,126,106,14,94, %T A058099 70,284,296,60 %V A058099 1,0,-3,6,2,2,-5,-16,12,2,17,-10,-48,56,10,24,-35,-126,106,14,94, %W A058099 -70,-284,296,60 %N A058099 McKay-Thompson series of class 10C for Monster. %D A058099 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994). %Y A058099 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc. %K A058099 sign,done %O A058099 -1,3 %A A058099 njas, Nov 27 2000 %I A021280 %S A021280 0,0,3,6,2,3,1,8,8,4,0,5,7,9,7,1,0,1,4,4,9,2,7,5,3,6,2,3,1,8,8,4,0, %T A021280 5,7,9,7,1,0,1,4,4,9,2,7,5,3,6,2,3,1,8,8,4,0,5,7,9,7,1,0,1,4,4,9,2, %U A021280 7,5,3,6,2,3,1,8,8,4,0,5,7,9,7,1,0,1,4,4,9,2,7,5,3,6,2,3,1,8,8,4,0 %N A021280 Decimal expansion of 1/276. %K A021280 nonn,cons %O A021280 0,3 %A A021280 njas %I A021738 %S A021738 0,0,1,3,6,2,3,9,7,8,2,0,1,6,3,4,8,7,7,3,8,4,1,9,6,1,8,5,2,8,6,1,0, %T A021738 3,5,4,2,2,3,4,3,3,2,4,2,5,0,6,8,1,1,9,8,9,1,0,0,8,1,7,4,3,8,6,9,2, %U A021738 0,9,8,0,9,2,6,4,3,0,5,1,7,7,1,1,1,7,1,6,6,2,1,2,5,3,4,0,5,9,9,4,5 %N A021738 Decimal expansion of 1/734. %K A021738 nonn,cons %O A021738 0,4 %A A021738 njas %I A016614 %S A016614 3,6,2,4,3,4,0,9,3,2,9,7,6,3,6,5,1,3,1,1,7,9,5,3,1,7,8,1,9,1,6,7,2, %T A016614 4,4,1,5,6,2,3,1,9,3,1,3,1,9,9,9,5,2,9,6,4,1,4,3,9,3,1,0,1,0,7,0,9, %U A016614 2,4,5,8,6,4,6,6,6,4,2,2,9,7,8,0,5,2,8,0,2,0,1,8,4,0,0,3,6,7,2,6,2 %N A016614 Decimal expansion of ln(75/2). %K A016614 nonn,cons %O A016614 1,1 %A A016614 njas %I A019769 %S A019769 3,6,2,4,3,7,5,7,7,1,2,7,8,7,2,6,9,8,0,4,8,0,3,8,3,2,9,5,1,3,6,8,8, %T A019769 3,3,3,0,3,4,2,9,9,6,1,2,4,9,3,3,2,7,9,4,3,3,2,8,9,2,9,0,1,7,0,2,9, %U A019769 8,7,6,8,8,4,0,4,7,1,3,9,6,7,9,2,7,6,1,8,4,2,9,0,4,7,0,0,2,2,1,9,0 %N A019769 Decimal expansion of 2*E/15. %K A019769 nonn,cons %O A019769 0,1 %A A019769 njas %I A021969 %S A021969 0,0,1,0,3,6,2,6,9,4,3,0,0,5,1,8,1,3,4,7,1,5,0,2,5,9,0,6,7,3,5,7,5, %T A021969 1,2,9,5,3,3,6,7,8,7,5,6,4,7,6,6,8,3,9,3,7,8,2,3,8,3,4,1,9,6,8,9,1, %U A021969 1,9,1,7,0,9,8,4,4,5,5,9,5,8,5,4,9,2,2,2,7,9,7,9,2,7,4,6,1,1,3,9,8 %N A021969 Decimal expansion of 1/965. %K A021969 nonn,cons %O A021969 0,5 %A A021969 njas %I A046901 %S A046901 1,3,6,2,7,1,8,16,7,17,6,18,5,19,4,20,3,21,2,22,1,23,46,22,47,21, %T A046901 48,20,49,19,50,18,51,17,52,16,53,15,54,14,55,13,56,12,57,11,58, %U A046901 10,59,9,60,8,61,7,62,6,63,5,64,4,65,3,66,2,67,1,68,136,67,137 %N A046901 a(n) = a(n-1)-n if a(n-1)>n, else a(n) = a(n-1)+n. %C A046901 Variation (1) on Recaman's sequence A005132. Variation (4) (A064389) is the nicest, after A005132 itself. %H A046901 Index entries for sequences related to Recaman's sequence %F A046901 This is a concatenation S_0, S_1, S_2, ... where S_i = [b_0, b_1, ..., b_{k-1}], k=5*3^i, with b_0 = 1, b_{2j} = k+j, b_{2j+1} = (k+1)/2-j. E.g. S_0 = [1,3,6,2,7]. %p A046901 A046901:=proc(n) option remember; if n = 1 then 1 else if A046901(n-1)>n then A046901(n-1)-n else A046901(n-1)+n; fi; fi; end; %Y A046901 Cf. A008344, A005132. %K A046901 easy,nonn,nice %O A046901 1,2 %A A046901 njas %I A065232 %S A065232 1,1,3,6,2,7,13,20,7,16,26,37,9,22,36,51,2,19,37,56,76,1,23,46,70,95, %T A065232 121,15,43,72,102,133,165,22,56,91,127,164,202,16,56,97,139,182,226, %U A065232 271,37,84,132,181,231,282,334,46,100,155,211,268,326,385,37,98,160 %N A065232 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the octagonal numbers. The first elements of the rows form a(n). %Y A065232 Cf. A064766, A064865, A065221-A065231, A065233-A065234. %K A065232 easy,nonn %O A065232 0,3 %A A065232 Floor van Lamoen (f.v.lamoen@wxs.nl), Oct 22 2001 %I A005132 M2511 %S A005132 1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,8,25,43,62,42,63,41,18, %T A005132 42,17,43,16,44,15,45,14,46,79,113,78,114,77,39,78,38,79,37,80, %U A005132 36,81,35,82,34,83,33,84,32,85,31,86,30,87,29,88,28,89,27,90,26 %N A005132 Recaman's sequence: a(n) = a(n-1)-n if positive and new, else a(n) = a(n-1)+n. %C A005132 I conjecture that every number eventually appears - see A057167, A064227, A064228. - njas. %H A005132 C. L. Mallows, Plot (jpeg) of first 10000 terms %H A005132 C. L. Mallows, Plot (postscript) of first 10000 terms %H A005132 N. J. A. Sloane, My favorite integer sequences (Abtract, pdf, ps), in Sequences and their Applications (Proceedings of SETA '98), C. Ding, T. Helleseth and H. Niederreiter (editors), Springer-Verlag, London, 1999, pp. 103-130. %H A005132 N. J. A. Sloane, Fortran program for A005132, A057167, A064227, A064228 %H A005132 Index entries for sequences related to Recaman's sequence %e A005132 Consider n=6. We have a(5)=7 and try to subtract 6. The result, 1, is certainly positive, but we cannot use it because 1 is already in the sequence. So we must add 6 instead, getting a(6) = 7 + 6 = 13. %p A005132 h:= array(1..100000); maxt:=100000; a:=[1]; ad:=[1]; su:=[]; h[1]:=1; for nx from 2 to 500 do t1:=a[nx-1]-nx; if t1>0 and h[t1] <> 1 then su:=[op(su),nx]; else t1:=a[nx-1]+nx; ad:=[op(ad),nx]; fi; a:=[op(a),t1]; if t1 <= maxt then h[t1]:= 1; fi; od: # a is A005132, ad is A057165, su is A057166 %t A005132 a = {1}; Do[ If[ a[ [ -1 ] ] - n > 0 && Position[ a, a[ [ -1 ] ] - n ] == {}, a = Append[ a, a[ [ -1 ] ] - n ], a = Append[ a, a[ [ -1 ] ] + n ] ], {n, 2, 70} ]; a %Y A005132 Cf. A057165 (addition steps), A057166 (subtraction steps), A057167 (steps to hit n), A008336, A046901 (simplified version), A063733. %Y A005132 Cf. A064227 (records for reaching n), A064228 (n's that take a record number of steps to reach), A064284 (no. of times n appears), A064288, A064289, A064290 (heights of terms). %Y A005132 Cf. A064291 (record highs), A064387, A064388, A064389 (further variants). %Y A005132 A row of A066201. %K A005132 easy,nonn,nice %O A005132 1,2 %A A005132 B. Recaman [Recam\'{a}n], njas, sp %E A005132 Allan Wilks (allan@research.att.com), Nov 06, 2001, computed 10^15 terms of this sequence. At this point the smallest missing number is 852655. %I A064388 %S A064388 1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,8,25,43,62,42,63,41,18,44,17, %T A064388 45,16,46,15,47,14 %N A064388 Variation (3) on Recaman's sequence (A005132): set s (the step size) initially equal to 2; to get a(n), we first try to subtract s from a(n-1): a(n) = a(n-1)-s if positive and not already in the sequence, and change s to s+1; if not, a(n) = a(n-1)+s+i, where i >= 0 is the smallest number such that a(n-1)+s+i has not already appeared, and change s to s+i+1 %C A064388 Variation (4) (A064389) is the nicest of these variations. %C A064388 I would also like to get the following sequences: number of steps before n appears (or 0 if n never appears), list of numbers that never appear, height of n (cf. A064288, A064289, A064290), etc. %H A064388 Index entries for sequences related to Recaman's sequence %Y A064388 Cf. A005132, A046901, A064387, A064389. %K A064388 nonn,easy,more %O A064388 1,2 %A A064388 njas, Sep 28 2001 %I A064387 %S A064387 1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,8,25,43,62,42,63,41,18, %T A064387 44,19,45,72,100,71,101,70,38,5,39,4,40,77,115,76,36,78,120,163, %U A064387 119,74,28,75,27,79,29,80,132,185,131,186,130,73,15,81,141,202 %N A064387 Variation (2) on Recaman's sequence (A005132): to get a(n), we first try to subtract n from a(n-1): a(n) = a(n-1)-n if positive and not already in the sequence; if not then a(n) = a(n-1)+n+i, where i >= 0 is the smallest number such that a(n-1)+n+i has not already appeared. %C A064387 Variation (4) (A064389) is the nicest of these variations. %C A064387 I would also like to get the following sequences: number of steps before n appears (or 0 if n never appears), list of numbers that never appear, height of n (cf. A064288, A064289, A064290), etc. %D A064387 Suggested by J. C. Lagarias. %H A064387 Index entries for sequences related to Recaman's sequence %p A064387 h:= array(1..100000); maxt:=100000; a:= array(1..1000); a[1]:=1; h[1]:=1; for nx from 2 to 1000 do t1:=a[nx-1]-nx; if t1>0 and h[t1] <> 1 then a[nx]:=t1; if t1 < maxt then h[t1]:=1; fi; else for i from 0 to 1000 do t1:=a[nx-1]+nx+i; if h[t1] <> 1 then a[nx]:=t1; if t1 < maxt then h[t1]:=1; fi; break; fi; od; fi; od; evalm(a); %Y A064387 Cf. A005132, A046901, A064388, A064389. Agrees with A064389 for first 187 terms, then diverges. %K A064387 nonn,easy %O A064387 1,2 %A A064387 njas, Sep 28 2001 %I A064389 %S A064389 1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,8,25,43,62,42,63,41,18, %T A064389 44,19,45,72,100,71,101,70,38,5,39,4,40,77,115,76,36,78,120,163, %U A064389 119,74,28,75,27,79,29,80,132,185,131,186,130,73,15,81,141,202 %N A064389 Variation (4) on Recaman's sequence (A005132): to get a(n), we first try to subtract n from a(n-1): a(n) = a(n-1)-n if positive and not already in the sequence; if not then we try to add n: a(n) = a(n-1)+n if not already in the sequence; if this fails we try to subtract n+1 from a(n-1), or to add n+1 to a(n-1), or to subtract n+2, or to add n+2, etc., until one of these produces a positive number not already in the sequence - this is a(n). %C A064389 This is the nicest of these variations. Is this a permutation of the natural numbers? %C A064389 I would also like to get the following sequences: number of steps before n appears (or 0 if n never appears), list of numbers that never appear (if any), height of n (cf. A064288, A064289, A064290), etc. %D A064389 Suggested by J. C. Lagarias. %H A064389 Index entries for sequences related to Recaman's sequence %H A064389 Index entries for sequences that are permutations of the natural numbers %p A064389 h:= array(1..100000); maxt:=100000; a:= array(1..1000); a[1]:=1; h[1]:=1; for nx from 2 to 1000 do for i from 0 to 100 do t1:=a[nx-1]-nx-i; if t1>0 and h[t1] <> 1 then a[nx]:=t1; if t1 < maxt then h[t1]:=1; fi; break; fi; t1:=a[nx-1]+nx+i; if h[t1] <> 1 then a[nx]:=t1; if t1 < maxt then h[t1]:=1; fi; break; fi; od; od; evalm(a); %Y A064389 Cf. A005132, A046901, A064387, A064388. Agrees with A064387 for first 187 terms, then diverges. %K A064389 nonn,easy,nice %O A064389 1,2 %A A064389 njas, Sep 28 2001 %I A002516 %S A002516 0,3,6,2,12,7,4,10,24,11,14,18,8,15,20,26,48,19,22,34,28,23,36,42,16, %T A002516 27,30,50,40,31,52,58,96,35,38,66,44,39,68,74,56,43,46,82,72,47,84,90, %U A002516 32,51,54,98,60,55,100,106,80,59,62,114,104,63,116,122,192,67,70,130 %N A002516 Earliest sequence with a(a(n))=2n. %F A002516 a(4n) = 2*(a(2n)), a(4n+1) = 4n+3, a(4n+2) = 2*(a(2n+1)), a(4n+3) = 8n+2. %Y A002516 Cf. A002517, A007379. %K A002516 nonn,nice %O A002516 0,2 %A A002516 Colin Mallows, colinm@research.avayalabs.com %E A002516 Formula and more terms from Henry Bottomley (se16@btinternet.com), Apr 27 2000 %I A010619 %S A010619 3,6,3,4,2,4,1,1,8,5,6,6,4,2,7,9,3,1,7,7,8,2,4,2,3,5,1,2,6,5,4,5,2, %T A010619 1,0,0,4,8,5,6,4,2,0,9,2,6,2,8,2,4,3,9,3,4,2,9,6,2,6,6,8,5,9,5,8,6, %U A010619 2,6,1,9,4,7,8,9,1,8,6,0,3,7,3,1,2,9,4,2,8,3,4,0,8,2,5,2,8,3,4,1,4 %N A010619 Decimal expansion of cube root of 48. %K A010619 nonn,cons %O A010619 1,1 %A A010619 njas %I A019715 %S A019715 1,3,6,3,4,2,6,0,3,9,1,5,8,0,8,9,2,5,1,7,6,7,8,2,1,1,4,1,0,3,1,6,5, %T A019715 0,1,4,0,6,1,3,4,9,9,5,8,1,6,3,2,6,0,5,4,8,3,2,4,1,3,9,0,6,0,7,5,7, %U A019715 9,1,6,2,5,2,6,6,0,8,4,8,6,2,9,3,3,6,5,3,4,6,6,5,9,0,2,9,6,4,9,7,4 %N A019715 Decimal expansion of sqrt(Pi)/13. %K A019715 nonn,cons %O A019715 0,2 %A A019715 njas %I A038138 %S A038138 1,3,6,3,6,2,0,1,3,6,3,6,2,0,1,3,6,3,6,2,0,1,3,6,3,6,2,0,1,3,6,3,6,2,0, %T A038138 1,3,6,3,6,2,0,1,3,6,3,6,2,0,1,3,6,3,6,2,0,1,3,6,3,6,2,0,1,3,6,3,6,2,0, %U A038138 1,3,6,3,6,2,0,1,3,6,3,6,2,0,1,3,6,3,6,2,0,1,3,6,3,6,2,0,1,3,6,3,6,2,0 %N A038138 Order of n (mod 7). %K A038138 easy,nonn %O A038138 1,2 %A A038138 Felice Russo (felice.russo@katamail.com) %E A038138 More terms from Larry Reeves (larryr@acm.org), Apr 04 2000 %I A010704 %S A010704 3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3, %T A010704 6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6, %U A010704 3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3,6,3 %N A010704 Period 2. %K A010704 nonn %O A010704 0,1 %A A010704 njas %I A016661 %S A016661 3,6,3,7,5,8,6,1,5,9,7,2,6,3,8,5,7,6,9,4,2,6,2,5,9,5,5,3,3,4,6,0,3, %T A016661 0,1,0,5,3,1,2,8,7,9,3,9,5,6,5,9,3,8,4,0,7,2,6,5,8,6,4,0,2,4,5,9,0, %U A016661 2,6,8,6,3,2,4,0,3,3,8,9,2,0,0,4,3,2,0,0,2,1,7,1,0,8,1,2,1,5,8,5,8 %N A016661 Decimal expansion of ln(38). %D A016661 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2. %K A016661 nonn,cons %O A016661 1,1 %A A016661 njas %I A065231 %S A065231 1,1,3,6,3,8,14,3,11,20,30,7,19,32,46,6,22,39,57,76,15,36,58,81,105,18, %T A065231 44,71,99,128,10,41,73,106,140,175,22,59,97,136,176,217,24,67,111,156, %U A065231 202,249,11,60,110,161,213,266,320,33,89,146,204,263,323,384,43,106 %N A065231 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the heptagonal numbers. The first elements of the rows form a(n). %Y A065231 Cf. A064766, A064865, A065221-A065230, A065232-A065223. %K A065231 easy,nonn %O A065231 0,3 %A A065231 Floor van Lamoen (f.v.lamoen@wxs.nl), Oct 22 2001 %I A019918 %S A019918 3,6,3,9,7,0,2,3,4,2,6,6,2,0,2,3,6,1,3,5,1,0,4,7,8,8,2,7,7,6,8,3,4, %T A019918 0,4,3,8,9,0,4,7,1,7,8,3,7,5,3,7,3,8,1,1,4,1,9,5,6,1,2,9,8,8,7,1,3, %U A019918 0,7,3,9,6,2,1,0,0,4,8,9,6,3,8,8,2,4,3,8,5,4,5,7,4,0,3,1,4,6,3,8,5 %N A019918 Decimal expansion of tangent of 20 degrees. %K A019918 nonn,cons %O A019918 0,1 %A A019918 njas %I A055373 %S A055373 1,1,1,3,6,3,9,27,27,9,27,108,162,108,27,81,405,810,810,405,81,243, %T A055373 1458,3645,4860,3645,1458,243,729,5103,15309,25515,25515,15309,5103, %U A055373 729,2187,17496,61236,122472,153090,122472,61236,17496,2187,6561 %N A055373 Invert transform applied twice to Pascal's triangle A007318. %H A055373 N. J. A. Sloane, Transforms %H A055373 Index entries for triangles and arrays related to Pascal's triangle %F A055373 a(n,k)=3^(n-1)*C(n,k). %e A055373 1; 1,1; 3,6,3; 9,27,27,9; 27,108,162,108,27; ... %Y A055373 Cf. A000244, A007318, A055372, A055374. %K A055373 nonn,tabl %O A055373 0,4 %A A055373 Christian G. Bower (bowerc@usa.net), May 16 2000 %I A021737 %S A021737 0,0,1,3,6,4,2,5,6,4,8,0,2,1,8,2,8,1,0,3,6,8,3,4,9,2,4,9,6,5,8,9,3, %T A021737 5,8,7,9,9,4,5,4,2,9,7,4,0,7,9,1,2,6,8,7,5,8,5,2,6,6,0,3,0,0,1,3,6, %U A021737 4,2,5,6,4,8,0,2,1,8,2,8,1,0,3,6,8,3,4,9,2,4,9,6,5,8,9,3,5,8,7,9,9 %N A021737 Decimal expansion of 1/733. %K A021737 nonn,cons %O A021737 0,4 %A A021737 njas %I A011307 %S A011307 1,3,6,4,2,6,1,6,0,1,8,2,1,3,6,5,9,2,9,5,8,6,0,3,5,2,7,5,2,3,9,6,1, %T A011307 1,5,8,4,4,3,9,4,6,5,0,5,7,6,3,5,0,7,5,6,3,5,6,5,7,0,8,6,5,5,2,4,2, %U A011307 6,8,0,0,3,9,7,7,7,1,5,4,5,7,1,5,7,6,0,0,0,7,8,1,3,3,7,8,8,0,6,1,4 %N A011307 Decimal expansion of 8th root of 12. %K A011307 nonn,cons %O A011307 1,2 %A A011307 njas %I A006464 M2512 %S A006464 0,3,6,4,4,2,4,6,4,2,6,4,2,4,4,6,4,2,6,4,4,2,4,6,2,4,6,4,2,4,4,6,4,2,6, %T A006464 4,4,2,4,6,4,2,6,4,2,4,4,6,2,4,6,4,4,2,4,6,2,4,6,4,2,4,4,6,4,2,6,4,4,2, %U A006464 4,6,4,2,6,4,2,4,4,6,4,2,6,4,4,2,4,6,2,4,6,4,2,4,4,6,2,4,6,4,4,2,4,6,4 %N A006464 Continued fraction for Sum[ 1/4^(2^n),{n,0,Infinity} ]. %C A006464 A006464(n)=A004200(n) if n=0; A004200(n)+1 if n>0 (according to case u=3, b=1 of Theorem 5 (of the reference) which states that: if B(u,infinity)=Sum[ 1/u^(2^n),{n,0,Infinity} ]= [a0, a1, a2,... ] then B(u + b,infinity) = [a0, a1+b, a2+b, a3+b,... ] (u >= 3, b >= 0)). %D A006464 Shallit, Jeffrey; Simple continued fractions for some irrational numbers. J. Number Theory 11 (1979), no. 2, 209-217. %K A006464 nonn,cofr %O A006464 0,2 %A A006464 njas %E A006464 Better description and more terms from Antonio G. Astudillo (afg_astudillo@hotmail.com), Jun 19 2001 %I A023676 %S A023676 3,6,4,4,8,12,24,5,10,20,60,120,6,6,12,12,18,24,24,24,36,36,48,60, %T A023676 72,120,360,720,7,14,21,42,168,2520,5040 %N A023676 Table of orders of transitive permutation groups by degree. %D A023676 M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989. Appendix: Numerical Tables, p. 430. %H A023676 Index entries for sequences related to groups %e A023676 {3,6}, {4,4,8,12,24}, ... %Y A023676 Cf. A000001, A000019, A023675. %K A023676 nonn,tabf,nice %O A023676 3,1 %A A023676 njas %I A011287 %S A011287 1,1,3,6,4,6,3,6,6,6,3,8,5,7,2,4,7,4,6,6,3,7,5,6,2,9,6,7,3,2,4,7,6, %T A011287 2,4,0,8,7,7,4,0,8,2,2,5,3,2,3,2,0,9,0,7,2,7,6,4,2,2,2,2,2,7,3,8,2, %U A011287 6,6,6,3,2,9,3,9,1,1,9,5,4,8,0,6,7,5,7,3,0,5,8,7,5,2,4,4,2,9,9,4,8 %N A011287 Decimal expansion of 18th root of 10. %K A011287 nonn,cons %O A011287 1,3 %A A011287 njas %I A004546 %S A004546 1,3,6,4,8,5,8,0,5,5,7,8,6,1,5,3,0,3,6,0,8,1,7,4,6,6,8,1,6,7,3,8,3, %T A004546 3,0,8,4,7,1,5,7,2,2,5,3,2,2,0,7,6,5,1,1,8,5,5,5,5,5,6,6,7,8,1,2,5, %U A004546 7 %N A004546 Expansion of sqrt(2) in base 9. %K A004546 nonn,base,cons %O A004546 1,2 %A A004546 njas %I A021278 %S A021278 0,0,3,6,4,9,6,3,5,0,3,6,4,9,6,3,5,0,3,6,4,9,6,3,5,0,3,6,4,9,6,3,5, %T A021278 0,3,6,4,9,6,3,5,0,3,6,4,9,6,3,5,0,3,6,4,9,6,3,5,0,3,6,4,9,6,3,5,0, %U A021278 3,6,4,9,6,3,5,0,3,6,4,9,6,3,5,0,3,6,4,9,6,3,5,0,3,6,4,9,6,3,5,0,3 %N A021278 Decimal expansion of 1/274. %K A021278 nonn,cons %O A021278 0,3 %A A021278 njas %I A065230 %S A065230 1,1,3,6,4,9,15,7,15,24,6,17,29,42,11,26,42,59,11,30,50,71,2,25,49,74, %T A065230 100,7,35,64,94,125,4,37,71,106,142,179,27,66,106,147,189,1,45,90,136, %U A065230 183,231,4,54,105,157,210,264,319,50,107,165,224,284,345,29,92,156,221 %N A065230 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the hexagonal numbers. The first elements of the rows form a(n). %Y A065230 Cf. A064766, A064865, A065221-A065229, A065231-A065223. %K A065230 easy,nonn %O A065230 0,3 %A A065230 Floor van Lamoen (f.v.lamoen@wxs.nl), Oct 22 2001 %I A009782 %S A009782 0,1,3,6,4,45,339,1442,2064,33309,425935,2853246,6127188,142899809, %T A009782 2558171511,23368092810,66908903744,2040609258393,46968028923771, %U A009782 543448366366294,1945041831809220,73278278468928309 %V A009782 0,1,-3,6,-4,-45,339,-1442,2064,33309,-425935,2853246,-6127188,-142899809, %W A009782 2558171511,-23368092810,66908903744,2040609258393,-46968028923771, %X A009782 543448366366294,-1945041831809220,-73278278468928309 %N A009782 Expansion of tanh(ln(1+x))/exp(x). %t A009782 Tanh[ Log[ 1+x ] ]/Exp[ x ] %K A009782 sign,done,easy %O A009782 0,3 %A A009782 rhh@research.bell-labs.com %E A009782 Extended with signs 03/97 by Olivier Gerard. %I A016615 %S A016615 3,6,5,0,6,5,8,2,4,1,2,9,3,7,3,8,5,3,9,7,5,0,0,6,4,1,9,9,9,5,0,1,3, %T A016615 2,4,6,1,3,8,3,2,9,1,4,4,9,1,5,9,0,2,3,1,0,9,5,5,7,2,7,7,8,4,9,5,7, %U A016615 4,7,5,9,8,6,4,6,9,5,6,1,1,2,6,5,2,5,6,8,4,9,8,7,9,5,9,4,0,8,7,6,3 %N A016615 Decimal expansion of ln(77/2). %K A016615 nonn,cons %O A016615 1,1 %A A016615 njas %I A019652 %S A019652 3,6,5,2,8,5,2,9,5,6,6,5,2,8,4,7,3,3,0,6,7,7,0,2,8,8,4,5,1,3,4,6,0, %T A019652 2,8,9,4,2,4,7,3,8,4,4,4,0,8,0,8,2,0,8,5,7,3,7,7,2,1,7,5,0,1,4,3,3, %U A019652 9,9,4,5,2,1,5,4,5,3,6,9,6,1,0,5,5,5,4,9,4,5,7,0,9,7,0,8,8,2,3,6,5 %N A019652 Decimal expansion of sqrt(Pi*E)/8. %K A019652 nonn,cons %O A019652 0,1 %A A019652 njas %I A011223 %S A011223 1,1,3,6,5,3,3,4,7,6,0,0,9,7,2,4,3,2,0,1,7,8,0,6,0,1,1,0,6,3,8,8,1, %T A011223 4,3,7,0,8,6,9,0,8,5,1,0,0,1,9,6,8,9,5,2,2,3,2,0,2,0,3,9,1,2,3,2,5, %U A011223 6,3,3,2,3,5,4,5,4,3,1,8,1,0,9,1,7,6,7,0,4,0,0,5,6,9,8,8,2,9,7,4,5 %N A011223 Decimal expansion of 14th root of 6. %K A011223 nonn,cons %O A011223 1,3 %A A011223 njas %I A063520 %S A063520 1,3,6,5,8,8,8,14,13,9,14,17,8,18,23,18,14,17,13,33,23,10,19,36,15,22,32, %T A063520 22,19,26,17,39,24,18,50,45,8,22,39,38,22,27,13,50,45,16,27,52,24,39,38, %U A063520 27,20,50,45,72,24,12,31,58,15,28,69,45,49,39,12,52,40,33,33,66,12,33,64 %N A063520 Sum divides product: number of solutions (r,s,t), r>=s>=t>0, to the equation rst = n(r+s+t). %C A063520 Number of solutions (r,s) in positive integers to the equation rs = n(r+s) is tau(n^2), cf. A048691. Number of solutions (r,s), r>=s>0, to the equation rs = n(r+s) is (tau(n^2)+1)/2, cf. A018892. %C A063520 Conjecturally, includes all positive integers except 2, 4, 7, and 11 - dww %D A063520 M. J. Pelling, "The Sum Divides the Product", Problem 10745, Amer. Math. Monthly, vol. 106 (1999), p. 587; vol 108, (no. 7, Aug. 2001), pp. 668-669. [Gives upper bound] %e A063520 There are 8 such solutions to rst = 5(r+s+t): (5, 4, 3), (7, 5, 2), (10, 4, 2), (11, 10, 1), (15, 8, 1), (20, 7, 1), (25, 3, 2), (35, 6, 1). %Y A063520 Cf. A018892, A004194, A063525. %Y A063520 More terms from dww, Aug 01, 2001 %K A063520 nonn %O A063520 1,2 %A A063520 Jud McCranie (jud.mccranie@mindspring.com) and Vladeta Jovovic (vladeta@Eunet.yu), Aug 01 2001 %I A059770 %S A059770 0,3,6,5,8,17,7,12,32,9,25,14,38,51,16,31,46,13,57,52,20,15,85,99,22, %T A059770 60,110,96,132,66,120,26,167,19,79,137,53,97,188,206,21,30,80,203,187, %U A059770 91,157,249,201,34,142,166,222,194,296,94,67,36,283,324,27,102,113,73 %N A059770 First solution of x^2 = 2 mod p for primes p such that a solution exists. %C A059770 Solutions mod p are represented by integers from 0 to p-1. For p > 2: If x^2 = 2 has a solution mod p, then it has exactly two solutions and their sum is p; i is a solution mod p of x^2 = 2 iff p-i is a solution mod p of x^2 = 2. No integer occurs more than once in this sequence. Moreover, no integer (except 0) occurs both in this sequence and in sequence A059771 of the second solutions (Cf. A059772). %F A059770 a(n) = first (least) solution of x^2 = 2 mod p, where p is the n-th prime such that x^2 = 2 mod p has a solution, i.e. p is the n-th term of A038873. %e A059770 a(6) = 17, since 41 is the sixth term of A038873, 17 and 24 are the solutions mod 41 of x^2 = 2, and 17 is the smaller one. %Y A059770 Cf. A038873, A059771, A059772. %K A059770 nonn %O A059770 1,2 %A A059770 Klaus Brockhaus (klaus-brockhaus@t-online.de), Feb 21 2001 %I A019690 %S A019690 1,3,6,5,9,0,9,8,4,9,3,8,6,8,6,6,6,2,5,4,1,8,5,4,0,6,0,1,4,2,5,8,7, %T A019690 0,8,1,9,2,1,6,1,6,0,6,0,8,4,2,3,9,5,9,0,5,2,5,9,7,8,0,1,9,9,6,6,5, %U A019690 5,5,7,2,3,5,0,5,5,9,2,2,1,3,0,3,7,5,1,3,1,9,4,8,9,2,7,9,1,8,1,3,3 %N A019690 Decimal expansion of Pi/23. %K A019690 nonn,cons %O A019690 0,2 %A A019690 njas %I A010620 %S A010620 3,6,5,9,3,0,5,7,1,0,0,2,2,9,7,1,5,1,7,2,3,8,0,7,3,3,1,0,1,1,9,4,0, %T A010620 8,2,6,3,4,8,7,1,0,3,6,6,8,8,4,3,3,2,4,0,3,3,1,7,8,6,3,6,4,6,7,0,1, %U A010620 2,0,6,3,7,5,6,5,9,4,6,8,7,1,2,3,0,9,3,6,4,3,8,5,0,7,8,5,3,7,4,5,8 %N A010620 Decimal expansion of cube root of 49. %K A010620 nonn,cons %O A010620 1,1 %A A010620 njas %I A046128 %S A046128 3,6,5,9,4,13,9,8,11,7,12,10,13,12,15,7,10,3,17,17,20,6,18,11,8,26,5, %T A046128 18,16,21,8,15,25,19,15,13,12,16,17,25,10,15,22,14,24,13,25,15,9,17, %U A046128 20,26,15,17,27,12,13,39,29,24,30,21,39,14,27,26,20,25,13,24,27,25,37 %N A046128 Smallest side a of a scalene integer Heronian triangles sorted by increasing c and b. %H A046128 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A046128 Cf. A046129, A046130, A046131. %K A046128 nonn %O A046128 0,1 %A A046128 Eric W. Weisstein (eric@weisstein.com) %I A057098 %S A057098 3,6,5,9,8,12,7,10,15,20,18,9,12,16,21,15,24,14,11,27,20,24,30,16,28, %T A057098 33,13,40,25,36,21,18,33,24,32,39,42,30,15,48,20,45,36,48,40,35,28,39, %U A057098 51,22,60,54,17,27,40,57,36,48,65,60,24,32,35,56,63,45,60,19,66,44,56 %N A057098 Shortest side of a Pythagorean triangle (ordered by the product of the sides). %F A057098 a(n) =A057096(n)/(A057099(n)*A057100(n)) =sqrt(A057100(n)^2-A057099(n)^2) %e A057098 a(1)=3 since 3*4*5=60 is smallest possible positive product %Y A057098 Cf. A009004, A009005, A046083, A057096. %K A057098 nonn %O A057098 1,1 %A A057098 Henry Bottomley (se16@btinternet.com), Aug 01 2000 %I A053628 %S A053628 3,6,5,9,8,12,7,10,20,18,16,12,9,24,27,14,20,28,11,40,16,25,33,39,32, %T A053628 21,48,18,13,48,36,40,51,39,35,54,20,28,57,65,15,36,60,22,27,69,35,44, %U A053628 32,50,17,66,24,64,81,55,42,51,87,100,96,36,26,93,85,84,19,96,52,49,33 %N A053628 Smallest integer which is the harmonic mean of A005279(n) and an integer. %C A053628 If m is not in A005279 then smallest harmonic mean is m. %Y A053628 Cf. A005279. %K A053628 nonn %O A053628 1,1 %A A053628 Henry Bottomley (se16@btinternet.com), Mar 20 2000 %I A046083 %S A046083 3,6,5,9,8,12,15,7,10,20,18,16,21,12,15,24,9,27,30,14,24,20,28,33,40, %T A046083 36,11,39,33,25,16,32,42,48,24,45,21,30,48,18,51,40,36,13,60,39,54,35, %U A046083 57,65,60,28,20,48,40,63,56,60,66,36,15,69,80,45,56,72,22,27,75,44,35 %N A046083 The smallest member 'a' of the Pythagorean triplets (a,b,c) ordered by increasing c. %H A046083 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A046083 Cf. A046084, A009000. %K A046083 nonn %O A046083 1,1 %A A046083 Eric W. Weisstein (eric@weisstein.com) %I A007479 M2513 %S A007479 0,3,6,5,12,2,10,9,24,11,4,14,20,15,18,17,48,26,22,21,8,23,28,38,40, %T A007479 27,30,29,36,50,34,33,96,35,52,62,44,39,42,41,16,74,46,45,56,47,76,86, %U A007479 51,54,53,60,98,58,57,72,59,100,110,68,63,66,65 %N A007479 Earliest sequence with a(a(a(n))) = 2n. %K A007479 easy,nonn,nice %O A007479 0,2 %A A007479 njas %I A048724 %S A048724 0,3,6,5,12,15,10,9,24,27,30,29,20,23,18,17,48,51,54,53,60,63,58,57,40, %T A048724 43,46,45,36,39,34,33,96,99,102,101,108,111,106,105,120,123,126,125, %U A048724 116,119,114,113,80,83,86,85,92,95,90,89,72,75,78,77,68,71,66,65,192 %N A048724 Write n and 2n in binary and add them mod 2. %C A048724 Reversing binary representation of -n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives -n. Note that the alternation is applied only to the non-zero bits and does not depend on the exponent of two. All integers have a unique reversing binary representation (see cited exercise for proof). Complement of A065621. - Marc Le Brun (mlb@well.com), Nov 07 2001 %C A048724 A permutation of the "evil" numbers A001969 - Marc Le Brun (mlb@well.com), Nov 07 2001 %D A048724 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 178, (exercise 4.1. Nr. 27) %F A048724 a(n) = Xmult(n,3) (or n XOR (n<<1)). a(n) = A065621(-n). %e A048724 12 = 1100 in binary, 24=11000, and their sum is 10100=20, so a(12)=20. %e A048724 a(4) = 12 = + 8 + 4 --> - 8 + 4 = -4. %Y A048724 Cf. A048720, A048725, A048726, A048728. Bisection of A003188. %Y A048724 See also A065620, A065621. %K A048724 nonn,nice,easy %O A048724 0,2 %A A048724 Antti.Karttunen@iki.fi (karttu@megabaud.fi), Apr 26, 1999 %I A009193 %S A009193 1,1,0,3,6,5,135,154,1260,6189,33285,470294,2056758,14299935,3614611, %T A009193 379101878,10041282472,121491285111,1656259499511,21619048472474, %U A009193 338632378045750,5874571029856693,115201962050532137 %V A009193 1,1,0,3,6,-5,135,-154,1260,6189,-33285,470294,-2056758,14299935,-3614611, %W A009193 -379101878,10041282472,-121491285111,1656259499511,-21619048472474, %X A009193 338632378045750,-5874571029856693,115201962050532137 %N A009193 Expansion of exp(ln(1+x).cosh(x)). %t A009193 Exp[ Log[ 1+x ]*Cosh[ x ] ] %K A009193 sign,done,easy %O A009193 0,4 %A A009193 rhh@research.bell-labs.com %E A009193 Extended with signs 03/97 by Olivier Gerard. %I A021736 %S A021736 0,0,1,3,6,6,1,2,0,2,1,8,5,7,9,2,3,4,9,7,2,6,7,7,5,9,5,6,2,8,4,1,5, %T A021736 3,0,0,5,4,6,4,4,8,0,8,7,4,3,1,6,9,3,9,8,9,0,7,1,0,3,8,2,5,1,3,6,6, %U A021736 1,2,0,2,1,8,5,7,9,2,3,4,9,7,2,6,7,7,5,9,5,6,2,8,4,1,5,3,0,0,5,4,6 %N A021736 Decimal expansion of 1/732. %K A021736 nonn,cons %O A021736 0,4 %A A021736 njas %I A021277 %S A021277 0,0,3,6,6,3,0,0,3,6,6,3,0,0,3,6,6,3,0,0,3,6,6,3,0,0,3,6,6,3,0,0,3, %T A021277 6,6,3,0,0,3,6,6,3,0,0,3,6,6,3,0,0,3,6,6,3,0,0,3,6,6,3,0,0,3,6,6,3, %U A021277 0,0,3,6,6,3,0,0,3,6,6,3,0,0,3,6,6,3,0,0,3,6,6,3,0,0,3,6,6,3,0,0,3 %N A021277 Decimal expansion of 1/273. %K A021277 nonn,cons %O A021277 0,3 %A A021277 njas %I A016662 %S A016662 3,6,6,3,5,6,1,6,4,6,1,2,9,6,4,6,4,2,7,4,4,8,7,3,2,6,7,8,4,8,7,8,4, %T A016662 4,3,0,9,4,5,2,7,5,8,5,0,2,5,8,2,9,5,6,5,6,8,1,5,3,7,3,9,8,4,4,3,0, %U A016662 0,9,5,8,9,6,0,5,4,3,0,1,9,1,4,6,2,7,3,1,9,0,4,1,8,2,5,4,2,2,1,5,7 %N A016662 Decimal expansion of ln(39). %D A016662 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2. %K A016662 nonn,cons %O A016662 1,1 %A A016662 njas %I A040006 %S A040006 3,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, %T A040006 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, %U A040006 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6 %N A040006 Continued fraction for sqrt(10). %H A040006 Index entries for continued fractions for constants %p A040006 Digits:=100: convert(evalf(sqrt(N)),confrac,90,'cvgts'): %K A040006 nonn,cofr,easy %O A040006 0,1 %A A040006 njas %I A025500 %S A025500 1,1,3,6,6,8,7,8,13,12,14,11,16,18,18,21,20,21,19,21,25,24,29,31,27,32, %T A025500 32,34,36,37,33,39,44,43,35,37,43,44,56,45,46,48,49,57,55,54,53,54,54,66, %U A025500 62,65,61,63,60,69,65,68,67,66,66,74,75,77,82,74,82,77,69,82,82,92,83 %N A025500 Number of terms in Zeckendorf representation of 8^n. %Y A025500 Cf. A007895. %K A025500 nonn %O A025500 0,3 %A A025500 Clark Kimberling (ck6@cedar.evansville.edu) %I A002853 M2514 N0994 %S A002853 1,3,6,6,10,16,28,28,28,28,28,28,28,28,36,40,48,48 %N A002853 Maximal size of a set of equiangular lines in n dimensions. %D A002853 W. W. R. Ball and H. S. M. Coxeter,"Mathematical Recreations and Essays," 13th Ed. Dover, p. 307. %D A002853 P. W. H. Lemmens and J. J. Seidel, Equiangular lines, J. Algebra, 24 (1973), 494-512. %D A002853 F. Buekenhout, ed., Handbook of Incidence Geometry, 1995, p. 884. %K A002853 hard,nonn,nice %O A002853 1,2 %A A002853 njas %E A002853 The sequence continues: 72 <= a(19) <= 76, 90 <= a(20) <= 96, a(21) = 126, a(22) = 176, a(23) = ... = a(42) = 276, a(43) = 344. %I A012212 %S A012212 3,6,6,12,12,12,12,18,18,24,36,48,60, %T A012212 84,102,138,186,246,306,378,486,630,816,1050, %U A012212 1350,1782,2328,2988,3870,5076,6624 %N A012212 Number of square-free palindromes over {0, 1, 2} of length 2n+1. %K A012212 nonn,base %O A012212 0,1 %A A012212 Jeffrey Shallit (shallit@graceland.uwaterloo.ca) %I A036252 %S A036252 1,3,6,6,16,39,114,335,1081,3574,12408,44076,160915,598244,2263400, %T A036252 8681464,33713947,132305267,524095596,2093208435,8422013745, %U A036252 34110403728,138979989162,569339728312,2343898451275,9693334574919 %N A036252 Trees with 3-colored leaves. %H A036252 Index entries for sequences related to trees %F A036252 G.f.: A(x) = B(x)+B(x)^2/2+B(x^2)/2-B(x)*(B(x)-2*x), where B(x) = g.f. for A029857. %K A036252 nonn %O A036252 0,2 %A A036252 Christian G. Bower (bowerc@usa.net), Nov 1998. %I A032338 %S A032338 3,6,6,72,1440,26640,589680,13426560,353808000,9841305600, %T A032338 306920275200,10257340262400,375041008742400,14697998361446400, %U A032338 621176708984832000,28005677334835200000,1347486252598886400000 %N A032338 Identity bracelets with n labeled beads of 3 colors. %H A032338 C. G. Bower, Transforms (2) %F A032338 "DHJ" (bracelets, identity, labeled) transform of 3,0,0,0... %F A032338 n! * A032240. %K A032338 nonn,huge %O A032338 1,1 %A A032338 Christian G. Bower (bowerc@usa.net) %I A065269 %S A065269 1,3,6,7,2,12,14,15,4,13,5,24,26,28,30,31,8,29,9,27,10,25,11,48,50,52, %T A065269 54,56,58,60,62,63,16,61,17,59,18,57,19,55,20,53,21,51,22,49,23,96,98, %U A065269 100,102,104,106,108,110,112,114,116,118,120,122,124,126,127,32,125,33 %N A065269 Infinite binary tree inspired permutation of N: 1 -> 1, 11ab..yz -> 11ab..yz0, 10ab..y1 -> 10ab..y, 10ab..y0 -> 11AB..Y1 (where 1AB..Y1 is the complement of 0ab..y0). %C A065269 On the right side every node replaces its left child, on the left side the right children replace their parents, and the left children are reflected to the right side (becoming right children). See comment at A065263. %H A065269 Index entries for sequences that are permutations of the natural numbers %p A065269 LeftChildInverted := proc(n) local k; if(1 = n) then RETURN(1); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN(2*n); fi; if(1 = (n mod 2)) then RETURN((n-1)/2); fi; RETURN(2^(k+1) + ((2^(k+2))-1) - n); end; %Y A065269 A057114, A065263, A065275, A065281, A065287. Inverse: A065270, conjugated with A059893: A065271, and the inverse of that: A065272. %K A065269 nonn %O A065269 1,2 %A A065269 Antti.Karttunen@iki.fi Oct 28 2001 %I A055102 %S A055102 1,3,6,7,3,6,17,24,21,6,21,54,77,72,24,64,159,216,190,57,159, %T A055102 392,534,468,144,381,924,1220,1044,312,833,1992,2625,2244,669, %U A055102 1746,4138,5382,4530,1332,3474,8184,10591,8886,2607,6724,15711 %V A055102 1,-3,6,-7,3,6,-17,24,-21,6,21,-54,77,-72,24,64,-159,216,-190,57,159, %W A055102 -392,534,-468,144,381,-924,1220,-1044,312,833,-1992,2625,-2244,669, %X A055102 1746,-4138,5382,-4530,1332,3474,-8184,10591,-8886,2607,6724,-15711 %N A055102 Expansion of cube of continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/ ( 1+q^5/ ( 1+q^6/ ( 1+q^7/ ( 1+q^8/ ( 1+q^9/ ( 1+q^10/... ))))))))). %D A055102 G. E. Andrews, Simplicity and surprise in Ramanujan's "Lost" Notebook, Amer. Math. Monthly, 104 (No. 10, Dec. 1997), 918-925, %Y A055102 See A007325 for first power (which has an alternative g.f.), A055101 for square, A055103 for 4th power. %K A055102 sign,done,easy %O A055102 0,2 %A A055102 njas, Jun 14 2000 %E A055102 More terms from Kok Seng Chua (chuaks@ihpc.nus.edu.sg), Jun 20 2000 %I A016616 %S A016616 3,6,7,6,3,0,0,6,7,1,9,0,7,0,7,6,1,8,4,7,5,5,7,1,3,4,2,0,0,2,3,2,3, %T A016616 4,3,5,4,0,9,8,0,4,1,0,9,0,0,6,2,3,5,4,3,7,1,2,9,1,4,9,1,1,0,8,0,6, %U A016616 6,4,1,2,4,3,9,1,5,4,7,3,7,5,6,2,5,4,0,0,7,7,0,3,8,7,7,7,6,6,0,9,1 %N A016616 Decimal expansion of ln(79/2). %K A016616 nonn,cons %O A016616 1,1 %A A016616 njas %I A021276 %S A021276 0,0,3,6,7,6,4,7,0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7,0,5,8,8,2,3,5,2,9, %T A021276 4,1,1,7,6,4,7,0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7,0,5,8,8,2,3,5,2,9,4, %U A021276 1,1,7,6,4,7,0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7,0,5,8,8,2,3,5,2,9,4,1 %N A021276 Decimal expansion of 1/272. %K A021276 nonn,cons %O A021276 0,3 %A A021276 njas %I A003458 M2515 %S A003458 3,6,7,7,23,62,143,44,159,46,47,174,2239,239,719,241,5849, %T A003458 2098,2099,43196,14871,19574,35423,193049,2105,36287,1119, %U A003458 284,240479,58782,341087,371942,6459,69614,37619,152188,152189 %N A003458 All prime factors of C(a(n),n) exceed n. %D A003458 E. F. Ecklund, Jr., et al., A new function associated with the prime factors of C(n,k), Math. Comp., 28 (1974), 647-649. %D A003458 R. Scheidler and H. C. Williams, A method of tabulating the number-theoretic function g(k), Math. Comp., 59 (1992), 251-257. %K A003458 easy,nonn,nice %O A003458 1,1 %A A003458 njas, sp %I A048748 %S A048748 3,6,7,8,9,10,12,15,14,13,18,19,20,24,21,27,33,30,28,32,36,26,38,35,45, %T A048748 42,39,44,40,51,48,54,31,60,49,63,61,62,57,56,68,78,52,72,81,66,64,74, %U A048748 70,80,84,96,76,65,99,90,105,98,88,114,102,93,104,108,123,110,126,100 %N A048748 Mean integral divisors associated with A048747. %e A048748 For a(3)=7, n=20, and sum of divisors of 20 is 42, number of divisors is 6, so integral quotient is 7 (42/6) %K A048748 easy,nonn %O A048748 3,1 %A A048748 Enoch Haga (EnochHaga@msn.com) %I A030781 %S A030781 3,6,7,8,9,10,12,15,18,21,24,27,30,32,34,37,41,44,50,54,58,62, %T A030781 65,68,72,78,82,85 %N A030781 n-th term s(k) such that s(k)>s(j) for j=1,2,...,k-1, where s=A030777 (with s(0)=0). %K A030781 nonn %O A030781 1,1 %A A030781 Clark Kimberling, ck6@cedar.evansville.edu %I A047559 %S A047559 0,1,3,6,7,8,9,11,14,15,16,17,19,22,23,24,25,27,30,31,32,33,35,38,39, %T A047559 40,41,43,46,47,48,49,51,54,55,56,57,59,62,63,64,65,67,70,71,72,73,75, %U A047559 78,79,80,81,83,86,87,88,89 %N A047559 Congruent to {0, 1, 3, 6, 7} mod 8. %K A047559 nonn %O A047559 0,3 %A A047559 njas %I A053030 %S A053030 3,6,7,8,9,12,14,15,16,18,20,21,23,24,27,28,30,32,33,35,36,39,40,41,42, %T A053030 43,45,46,47,48,49,51,52,54,55,56,57,60,63,64,66,67,68,69,70,72,75,77, %U A053030 78,80,81,82,83,84,86,87,88,90,91,92,93,94,95,96,98,99,100,102,103,104 %N A053030 Numbers with 2 zeros in Fibonacci numbers mod m. %C A053030 m is on this list iff m does not have 1 or 4 zeros in the Fibonacci sequence modulo m %H A053030 M. Renault, Fibonacci sequence modulo m %Y A053030 Cf. A001176. %K A053030 nonn %O A053030 0,1 %A A053030 Henry Bottomley (se16@btinternet.com), Feb 23 2000 %I A051205 %S A051205 0,1,3,6,7,8,9,13,15,16,19,22,24,27,33,35,37,40,46,48,54,55,61,63,72, %T A051205 73,78,80,81,88,91,94,97,99,112,115,117,118,120,135,141,142,143,144, %U A051205 157,160,163,166,168,169,171,175,187,193,195,198,208,214,216,222,224 %N A051205 Of the form x^2-3^y >= 0. %K A051205 nonn %O A051205 0,3 %A A051205 dww %I A047283 %S A047283 0,1,3,6,7,8,10,13,14,15,17,20,21,22,24,27,28,29,31,34,35,36,38,41,42, %T A047283 43,45,48,49,50,52,55,56,57,59,62,63,64,66,69,70,71,73,76,77,78,80,83, %U A047283 84,85,87,90,91,92,94,97 %N A047283 Congruent to {0, 1, 3, 6} mod 7. %K A047283 nonn %O A047283 0,3 %A A047283 njas %I A047557 %S A047557 0,3,6,7,8,11,14,15,16,19,22,23,24,27,30,31,32,35,38,39,40,43,46,47,48, %T A047557 51,54,55,56,59,62,63,64,67,70,71,72,75,78,79,80,83,86,87,88,91,94,95, %U A047557 96,99,102,103,104,107,110 %N A047557 Congruent to {0, 3, 6, 7} mod 8. %K A047557 nonn %O A047557 0,2 %A A047557 njas %I A034091 %S A034091 0,1,3,6,7,8,16,21,22,36,42,55,76,108,123,140,144,156,172,240,259,312, %T A034091 366,384,504,531,568,656,810,924,1032,1056,1140,1260,1356,1698,2040, %U A034091 2088,2216,2520,2644,3108,3474,3480,4272,4572,4844,5280,5304,5412,6840 %N A034091 Records for sum of proper divisors function. %Y A034091 Cf. A001065, A034090. %K A034091 nonn,nice %O A034091 1,3 %A A034091 njas, JIMMY (jhbubby@avana.net) %E A034091 More terms from Erich Friedman (erich.friedman@stetson.edu). %I A021735 %S A021735 0,0,1,3,6,7,9,8,9,0,5,6,0,8,7,5,5,1,2,9,9,5,8,9,6,0,3,2,8,3,1,7,3, %T A021735 7,3,4,6,1,0,1,2,3,1,1,9,0,1,5,0,4,7,8,7,9,6,1,6,9,6,3,0,6,4,2,9,5, %U A021735 4,8,5,6,3,6,1,1,4,9,1,1,0,8,0,7,1,1,3,5,4,3,0,9,1,6,5,5,2,6,6,7,5 %N A021735 Decimal expansion of 1/731. %K A021735 nonn,cons %O A021735 0,4 %A A021735 njas %I A005767 %S A005767 3,6,7,9,11,12,13,14,15,17,18,19,21,22,23,24,25,26,27,28,29,30,31,33, %T A005767 34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57 %N A005767 Solutions to n^2 = a^2 + b^2 + c^2 (a,b,c > 0). %K A005767 nonn %O A005767 1,1 %A A005767 njas, RALPH PETERSON (ralphp@LIBRARY.NRL.NAVY.MIL) %I A026415 %S A026415 1,3,6,7,9,11,12,16,17,19,22,24,25,26,27,32,33,35,36,38,41,44, %T A026415 45,48,49,50,53,54,55,58,60,61,62,67,70,71,74,75,76,80,84,85 %N A026415 a(n) = n-th number k such that s(k) is odd, where s = A026409. %K A026415 nonn %O A026415 1,2 %A A026415 Clark Kimberling, ck6@cedar.evansville.edu %I A026406 %S A026406 1,3,6,7,9,11,12,16,18,19,22,24,25,27,28,30,33,34,36,39,43,44, %T A026406 45,46,48,50,52,53,56,57,60,61,64,68,69,70,73,75,76,79,80,82 %N A026406 a(n) = n-th number k such that s(k) is odd, where s = A026400. %K A026406 nonn %O A026406 1,2 %A A026406 Clark Kimberling, ck6@cedar.evansville.edu %I A047558 %S A047558 1,3,6,7,9,11,14,15,17,19,22,23,25,27,30,31,33,35,38,39,41,43,46,47,49, %T A047558 51,54,55,57,59,62,63,65,67,70,71,73,75,78,79,81,83,86,87,89,91,94,95, %U A047558 97,99,102,103,105,107,110 %N A047558 Congruent to {1, 3, 6, 7} mod 8. %K A047558 nonn %O A047558 0,2 %A A047558 njas %I A047242 %S A047242 0,1,3,6,7,9,12,13,15,18,19,21,24,25,27,30,31,33,36,37,39,42,43,45,48, %T A047242 49,51,54,55,57,60,61,63,66,67,69,72,73,75,78,79,81,84,85,87,90,91,93, %U A047242 96,97,99,102,103,105,108 %N A047242 Congruent to {0, 1, 3} mod 6. %K A047242 nonn %O A047242 0,3 %A A047242 njas %I A026227 %S A026227 1,3,6,7,9,12,15,16,18,19,21,24,25,27,30,33,34,36,39,42,43,45, %T A026227 46,48,51,52,54,55,57,60,61,63,66,69,70,72,73,75,78,79,81,84, %U A026227 87,88,90,93,96,97,99,100,102,105,106,108,111,114,115,117,120 %N A026227 a(n) = (1/3)*(s(n) + 2), where s = A026226. %K A026227 nonn %O A026227 1,2 %A A026227 Clark Kimberling, ck6@cedar.evansville.edu %I A026232 %S A026232 3,6,7,9,12,15,16,18,19,21,24,25,27,30,33,34,36,39,42,43,45,46, %T A026232 48,51,52,54,55,57,60,61,63,66,69,70,72,73,75,78,79,81,84,87, %U A026232 88,90,93,96,97,99,100,102,105,106,108,111,114,115,117,120,123 %N A026232 a(n) = (1/3)*(s(n) + 1), where s = A026231. %K A026232 nonn %O A026232 1,1 %A A026232 Clark Kimberling, ck6@cedar.evansville.edu %I A061641 %S A061641 0,1,3,6,7,9,12,15,18,19,21,24,25,27,30,33,36,37,39,42,43,45,48,51,54, %T A061641 55,57,60,63,66,69,72,73,75,78,79,81,84,87,90,93,96,97,99,102,105,108, %U A061641 109,111,114,115,117,120,123,126,127,129,132,133,135,138,141,144,145 %N A061641 Let {f(k,N), k=0,1,2,...} denote the (3x+1)-sequence with starting value N; a(n) denotes the smallest positive integer which is not contained in the union of f(k,0),...,f(k,a(n-1)). %C A061641 In other words a(n) is the starting value of the next '3x+1'-sequences in the sense that a(n) is not a value in any sequence f(k,N) with N < a(n). %C A061641 f(0,N)=N, f(k+1,N)=f(k,N)/2 if f(k,N) is even and f(k+1,N)=3*f(k,N)+1 if f(k,N) is odd. %C A061641 For all n, a(n) mod 6 is 0, 1 or 3. I conjecture that a(n)/n -> C=constant for n->oo, where C=2.311... %H A061641 Index entries for sequences related to 3x+1 (or Collatz) problem %e A061641 a(1)=1 since Im(f(k,0))={0} for all k and so 1 is not a value of f(k,0). a(2)=3 since Im(f(k,0)) union Im(f(k,1))={0,1,2,4} and 3 is the smallest positive integer not contained in this set. %K A061641 nice,nonn %O A061641 0,3 %A A061641 Frederick Magata (fmagata@mi.uni-koeln.de), Jun 14 2001 %I A023982 %S A023982 3,6,7,9,13,13,17,17,17,19,21,20,27,27,28,26,30,29,30,34,36,33,38,36,38,43, %T A023982 40,40,46,44,48,43,46,47,48,47,51,53,55,53,57,57,55,57,59,58,63,58,62,63,64, %U A023982 67,73,66,69,68,67,73,73,74,79,80,78,71,77,76,77,79,84,79,87,82,87,89,86,89 %N A023982 Sum of exponents in prime-power factorization of multinomial coefficient M(5n;3n,n,n). %K A023982 nonn %O A023982 1,1 %A A023982 Clark Kimberling (ck6@cedar.evansville.edu) %I A039591 %S A039591 0,1,3,6,7,9,18,19,21,36,37,39,42,43,45,54,55,57,101,106,108,109,111, %T A039591 114,115,117,126,127,129,161,176,196,206,216,217,219,222,223,225,234, %U A039591 235,237,252,253,255,258,259,261,270,271,273,317,322,324,325,327,330 %N A039591 Representation in base 6 has same number of 2's, 4's and 5's. %K A039591 nonn,base,easy %O A039591 0,3 %A A039591 Olivier Gerard (ogerard@ext.jussieu.fr) %I A008912 %S A008912 1,3,6,7,10,12,15,18,19,21,25,27,28,33,36,37,42,45,46,48,52,55, %T A008912 57,60,61,63,66,69,73,75,78,82,87,88,90,91,96,102,105,106,108, %U A008912 111,117,118,120,123,126,127,133,135,136,141,144,145,147,150 %N A008912 Truncated triangular numbers (of form n*(n-3)/2 - k^2+k*n+1 for 1<=k<=n). %H A008912 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (Abstract, pdf, ps). %Y A008912 Cf. A008867. %K A008912 nonn %O A008912 1,2 %A A008912 njas,rkg %I A047281 %S A047281 0,3,6,7,10,13,14,17,20,21,24,27,28,31,34,35,38,41,42,45,48,49,52,55, %T A047281 56,59,62,63,66,69,70,73,76,77,80,83,84,87,90,91,94,97,98,101,104,105, %U A047281 108,111,112,115,118,119,122 %N A047281 Congruent to {0, 3, 6} mod 7. %K A047281 nonn %O A047281 0,2 %A A047281 njas %I A035000 %S A035000 1,3,6,7,10,15,16,19,21,28,30,36,45,50,51,55,66,77,78,90,91,105,112, %T A035000 120,126,136,141,153,156,161,171,190,210,231,253,266,275,276,300,325, %U A035000 351,352,357,378,393,406,414,435,442,465,496,504,528,546,561,595,615 %N A035000 Non-trivial trinomial coefficients. %D A035000 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78. %Y A035000 Cf. A027907. %K A035000 nonn %O A035000 0,2 %A A035000 Erich Friedman (erich.friedman@stetson.edu) %I A024412 %S A024412 1,3,6,7,10,15,21,25,28,31,36,45,55,63,65,66,78,90,91,105,120,127,136,140, %T A024412 153,171,190,210,231,253,255,266,276,300,301,325,350,351,378,435,462,465, %U A024412 496,511,528,561,595,630,666,703,741,750,780,820,861,903,946,966,990,1023 %N A024412 Ordered Stirling numbers s(n,k) of the second kind. %K A024412 nonn %O A024412 1,2 %A A024412 Clark Kimberling (ck6@cedar.evansville.edu) %I A028754 %S A028754 3,6,7,11,12,13,14,15,17,19,22,24,26,27,28,29,30,34,35, %T A028754 38 %N A028754 Nonsquares mod 41. %K A028754 nonn,fini,full %O A028754 0,1 %A A028754 njas %I A028795 %S A028795 3,6,7,11,12,13,14,15,17,19,22,24,26,27,28,29,30,34,35, %T A028795 38,41,44,47,48,52,53,54,55,56,58,60,63,65,67,68,69,70, %U A028795 71,75,76,79 %N A028795 Nonsquares mod 82. %K A028795 nonn,fini,full %O A028795 0,1 %A A028795 njas %I A004780 %S A004780 3,6,7,11,12,13,14,15,19,22,23,24,25,26,27,28,29,30,31,35,38,39,43, %T A004780 44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,67,70, %U A004780 71,75,76,77,78,79,83,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100 %N A004780 Binary expansion contains 2 adjacent 1's. %K A004780 nonn %O A004780 0,1 %A A004780 njas %I A051146 %S A051146 1,3,6,7,11,12,13,15,22,23,31,56,57,59,62,63,99,100,101,103,110,111, %T A051146 119,120,121,123,126,127,171,172,173,175,190,191,239,240,241,243,246, %U A051146 247,251,252,253,255,310,311,319,328,329,331,334,335,339,340,341,343 %N A051146 Sequence b(n) mentioned in A051145. %K A051146 nonn,easy,nice %O A051146 1,2 %A A051146 njas, emr %E A051146 More terms from Larry Reeves (larryr@acm.org), Oct 03 2000 %I A022544 %S A022544 3,6,7,11,12,14,15,19,21,22,23,24,27,28,30,31,33,35,38, %T A022544 39,42,43,44,46,47,48,51,54,55,56,57,59,60,62,63,66,67, %U A022544 69,70,71,75,76,77,78,79 %N A022544 Not the sum of 2 squares. %H A022544 Index entries for sequences related to sums of squares %H A022544 Steven Finch, Landau-Ramanujan Constant %Y A022544 Complement of A001481. %K A022544 nonn,nice %O A022544 0,1 %A A022544 njas %I A022550 %S A022550 3,6,7,11,13,14,15,18,19,20,21,22,23,29,30,32,34,35,38, %T A022550 39,40,41,42,45,46,47,48,51,53,54,55,56,58,59,60,61,62, %U A022550 66,67,69,70,71,74,75,77,78,79 %N A022550 Not the sum of a square and a nonnegative cube. %K A022550 nonn %O A022550 0,1 %A A022550 njas %I A047556 %S A047556 3,6,7,11,14,15,19,22,23,27,30,31,35,38,39,43,46,47,51,54,55,59,62,63, %T A047556 67,70,71,75,78,79,83,86,87,91,94,95,99,102,103,107,110,111,115,118, %U A047556 119,123,126,127,131,134,135 %N A047556 Congruent to {3, 6, 7} mod 8. %K A047556 nonn %O A047556 0,1 %A A047556 njas %I A015819 %S A015819 3,6,7,11,15,19,21,23,27,31,33,42,43,51,57,59,62,69,70,71,77,79,83, %T A015819 84,87,93,103,131,139,141,159,163,165,177,187,189,191,199,211,213, %U A015819 223,235,237,251,267,270,271,282,285,287,297,311,315,316,321,330 %N A015819 phi(n + 3) | sigma(n). %K A015819 nonn %O A015819 0,1 %A A015819 Robert G. Wilson v (rgwv@kspaint.com) %I A053478 %S A053478 1,3,6,7,12,9,16,15,18,17,28,19,32,23,30,31,48,27,46,35,40,39,62,39,60, %T A053478 45,54,47,76,45,76,63,68,65,74,55,92,65,78,71,112,61,104,79,84,85,132, %U A053478 79,110,85,114,91,144,81,126,95,112,105,164,91,152,107,118,127,144,101 %N A053478 Sum of iterates when A000010, Euler-Phi is iterated until fixed point 1. %C A053478 For n=2^w, the sum is 2^(w+1)-1. %F A053478 Sum[Nest[Phi,n,j],{j,1,x[n]}], where x[n] = number of steps to reach fixed value. %e A053478 If Phi is applied repeatedly to n = 91, the iterates {91,72,24,8,4,2,1} are obtained. Their sum is a(91)=91+72+24+8+4+2+1=202. %Y A053478 Cf. A000010, A010554, A049099, A049100, A049107, A049108. %K A053478 nonn %O A053478 1,2 %A A053478 Labos E. (labos@ana1.sote.hu), Jan 14 2000 %I A028802 %S A028802 3,6,7,12,13,14,15,19,23,24,26,27,28,29,30,31,33,35,37, %T A028802 38,41,43,46,48,51,52,54,56,58,59,60,61,62,63,65,66,70, %U A028802 74,75,76,77,82,83,86 %N A028802 Nonsquares mod 89. %K A028802 nonn,fini,full %O A028802 0,1 %A A028802 njas %I A004755 %S A004755 3,6,7,12,13,14,15,24,25,26,27,28,29,30,31,48,49,50,51,52,53,54,55, %T A004755 56,57,58,59,60,61,62,63,96,97,98,99,100,101,102,103,104,105,106, %U A004755 107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122 %N A004755 Binary expansion starts 11. %K A004755 nonn %O A004755 0,1 %A A004755 njas %I A004760 %S A004760 0,1,3,6,7,12,13,14,15,24,25,26,27,28,29,30,31,48,49,50,51,52,53,54, %T A004760 55,56,57,58,59,60,61,62,63,96,97,98,99,100,101,102,103,104,105,106, %U A004760 107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122 %N A004760 Binary expansion does not begin 10. %K A004760 nonn %O A004760 0,3 %A A004760 njas %I A032849 %S A032849 1,3,6,7,12,13,15,24,26,27,30,31,48,49,52,53,55,60,61,63,96,98, %T A032849 99,104,106,107,110,111,120,122,123,126,127,192,193,196,197,199, %U A032849 208,209,212,213,215,220,221,223,240,241,244,245,247,252,253 %N A032849 Base 2 representation SUM{d(i)*2^i: i=0,1,...,m) has d(m)<=d(m-1)>=d(m-2)<=... %K A032849 nonn,base %O A032849 1,2 %A A032849 Clark Kimberling, ck6@cedar.evansville.edu %I A038591 %S A038591 1,3,6,7,12,13,18,19,21,27,30,31,36,37,42,43,48,54,55,61, %T A038591 63,69,73,75,78,84,85,90,91,96,97,102,109,114,120,121,123, %U A038591 127,129,135,139,141,144,150,151,156,163,168,169,174,180 %N A038591 Sizes of clusters in hexagonal lattice A_2 with 3-fold symmetry. %C A038591 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. %H A038591 G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2 %Y A038591 Union of A038588 and A038590. %K A038591 nonn %O A038591 0,2 %A A038591 njas %I A028792 %S A028792 3,6,7,12,14,15,17,24,27,28,29,30,33,34,35,37,39,41,43, %T A028792 47,48,53,54,56,57,58,59,60,61,63,66,68,69,70,71,74,75, %U A028792 77,78 %N A028792 Nonsquares mod 79. %K A028792 nonn,fini,full %O A028792 0,1 %A A028792 njas %I A048717 %S A048717 0,3,6,7,12,14,15,24,28,30,31,48,51,56,60,62,63,96,99,102, %T A048717 103,112,115,120,124,126,127,192,195,198,199,204,206,207, %U A048717 224,227,230,231,240,243,248,252,254,255,384,387,390,391 %N A048717 Binary expansion matches ((0)*00(1*)11)*(0*). %C A048717 1-bits occur only in groups of two or more, separated from other such groups by at least two 0-bits. %C A048717 All terms satisfy rule150(n) = 3*n (see A048705 and A048710). %Y A048717 Superset of A048719. Cf. A048733. %K A048717 nonn %O A048717 0,2 %A A048717 Antti.Karttunen@iki.fi (karttu@megabaud.fi), 30.3.1999 %I A022434 %S A022434 1,3,6,7,12,14,18,20,23,26,29,32,35,38,41,45,47,51,53,57,59,63,65, %T A022434 69,71,75,77,81,83,86,89,92,95,98,101,104,107,110,113,116,119,122, %U A022434 125,128,131,134 %N A022434 a(n) = c(n-1) + c(n-3) where c is the sequence of numbers not in a. %K A022434 nonn %O A022434 0,2 %A A022434 Clark Kimberling (ck6@cedar.evansville.edu) %I A047705 %S A047705 1,3,6,7,12,15,23,24,26,31,32,35,36,37,41,42,49,51,52,53,55,56, %T A047705 59,60,64,65,69,70,72,75,76,79,80,82,85,86,93,94,97,99,102,106, %U A047705 109,110,112,122,123,125,126,128,130,133,137,141,143,145,150,153 %N A047705 First differences of A047699. %K A047705 nonn %O A047705 0,2 %A A047705 njas %I A033053 %S A033053 1,3,6,7,13,15,26,27,30,31,53,55,61,63,106,107,110,111,122,123, %T A033053 126,127,213,215,221,223,245,247,253,255,426,427,430,431,442, %U A033053 443,446,447,490,491,494,495,506,507,510,511,853,855,861,863 %N A033053 Base 2 representation SUM{d(i)*2^i: i=0,1,...,m} has odd d(i) for all odd i. %K A033053 nonn,base %O A033053 1,2 %A A033053 Clark Kimberling, ck6@cedar.evansville.edu %I A051218 %S A051218 0,1,3,6,7,13,19,24,33,40,45,48,49,54,87,97,118,147,166,174,192,199, %T A051218 222,243,262,279,285,294,307,318,327,334,339,342,343,376,423,465,552, %U A051218 637,678,685,720,801,880,894,931,957,1032,1105,1176,1182,1245,1312 %N A051218 Of the form 7^x-y^2 >= 0. %K A051218 nonn %O A051218 0,3 %A A051218 dww %I A064291 %S A064291 1,3,6,7,13,20,21,22,23,24,25,43,62,63,79,113,114,157,224,225,226, %T A064291 227,228,265,367,368,369,370,379,494,495,631,632,633,634,635,636, %U A064291 643,833,1090,1091,1092,1093,1094,1095,1096,1097,1098,1182,1183 %N A064291 Record high values in Recaman's sequence A005132. %H A064291 Index entries for sequences related to Recaman's sequence %Y A064291 Cf. A005132, A064288, A064289, A064292, A064293, A064294. %K A064291 nonn %O A064291 1,2 %A A064291 njas, Sep 25 2001 %I A037015 %S A037015 0,1,3,6,7,14,15,28,30,31,57,60,62,63,120,121,124,126,127,241,248,249, %T A037015 252,254,255,483,496,497,504,505,508,510,511,966,993,995,1008,1009, %U A037015 1016,1017,1020,1022,1987,1990,2016,2017,2019,2032,2033,2040,2041,2044 %N A037015 Reading binary expansion from right to left, run lengths strictly increase. %Y A037015 Cf. A037013-A037016. %K A037015 nonn,easy,base,nice %O A037015 0,3 %A A037015 njas %E A037015 More terms from Patrick De Geest (pdg@worldofnumbers.com), 02/99. %I A056055 %S A056055 3,6,7,14,17,28,58,59,83,86,87,89,97,118,167,197,228,281,313,316,339, %T A056055 367,379,383,456,458,469,529,541,543,569,577,587,593,607,618,626,629, %U A056055 647,669,673,677,678,683,687,701,709,719,722,727,729,767,771,772,778 %N A056055 Integers > 1 where the decimal expansion of 1/n contains n as a string (if 1/n is finite, trailing zeros don't count). %C A056055 The sequence is probably infinite, since long-period primes (cf. A006883) especially with high first digit are likely candidates, but is there a proof? Does any n with finite expansion of 1/n (i.e. n = 2^k * 5^m) occur? %H A056055 Index entries for sequences related to decimal expansion of 1/n %e A056055 118 is okay, since 1/118 = 0.00847457627118... contains "118", 100 is not in it, because 1/100 = 0.01 doesn't contain "100" (0.0100 doesn't count) %K A056055 nonn,base %O A056055 1,1 %A A056055 Ulrich Schimke (ulrich_schimke@idg.de) %I A056703 %S A056703 1,3,6,7,19,27,43,55,207,1311 %N A056703 Primes of the form 2*10^n + 9*R_n, where R_n are Repunits (A002275) of length n. %t A056703 Do[ If[ PrimeQ[ 2*10^n + (10^n-1)], Print[n]], {n, 0, 1500}] n=9439 also is prime. %K A056703 hard,nonn %O A056703 0,2 %A A056703 Robert G. Wilson v (RGWv@kspaint.com), Aug 10 2000 %I A050867 %S A050867 1,3,6,7,21,29,46,49,54,57,73,90,118,169,207,267,287,333,354,366,370, %T A050867 463,493,497,507,655,729,735,757,1054,1377,1770,2098,2886,3225,3611, %U A050867 3625,3657,4849,5706,5749,5771,7110,8395,12150,15658,16182,16693,17363 %N A050867 231*2^n-1 is prime. %H A050867 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A050867 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A050867 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %K A050867 hard,nonn %O A050867 0,2 %A A050867 njas, Dec 29 1999 %I A019248 %S A019248 0,0,3,6,7,21,44,107,274,655,1484,3919,10003,24765,62528,161536, %T A019248 413954 %N A019248 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite VET = VPI-8 [ Si17O34 ]. %D A019248 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997. %H A019248 G. Thimm, Cycle sequences of crystal structures %K A019248 nonn %O A019248 3,3 %A A019248 Georg Thimm (mgeorg@ntu.edu.sg) %I A019210 %S A019210 0,0,3,6,7,28,49,100,247,692,1445,3457,9167,22971,56256,147690, %T A019210 383299 %N A019210 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite NON = Nonasil-[ 4158 ] [ Si88O176 ] . 4 R. %D A019210 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997. %H A019210 G. Thimm, Cycle sequences of crystal structures %K A019210 nonn %O A019210 3,3 %A A019210 Georg Thimm (mgeorg@ntu.edu.sg) %I A019121 %S A019121 0,0,3,6,7,28,53,106,273,741,1593,3737,10000,24476,59099,153277, %T A019121 391122 %N A019121 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite EUO = EU-1 Nan [ AlnSi112-n O224 ]. %D A019121 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997. %H A019121 G. Thimm, Cycle sequences of crystal structures %K A019121 nonn %O A019121 3,3 %A A019121 Georg Thimm (mgeorg@ntu.edu.sg) %I A011334 %S A011334 1,1,3,6,8,3,4,2,9,7,8,5,5,5,9,8,5,8,4,9,9,7,1,5,2,0,7,8,4,4,7,3,4, %T A011334 7,4,9,8,9,3,0,3,2,8,1,9,8,8,5,0,7,1,1,4,1,7,5,6,3,1,2,4,4,0,2,6,5, %U A011334 4,3,7,9,7,5,8,1,5,4,9,0,8,2,4,0,6,5,1,1,1,9,8,7,6,7,3,6,7,2,7,9,7 %N A011334 Decimal expansion of 20th root of 13. %K A011334 nonn,cons %O A011334 1,3 %A A011334 njas %I A010621 %S A010621 3,6,8,4,0,3,1,4,9,8,6,4,0,3,8,6,6,0,5,7,7,9,8,2,2,8,3,3,5,7,9,8,0, %T A010621 7,2,2,1,9,1,7,2,5,8,1,1,7,4,3,8,1,8,2,6,6,9,2,5,6,4,6,1,4,9,4,5,7, %U A010621 7,2,3,5,9,7,5,5,0,8,7,7,7,0,4,2,7,8,1,5,0,5,4,8,4,8,6,5,6,0,1,9,7 %N A010621 Decimal expansion of cube root of 50. %K A010621 nonn,cons %O A010621 1,1 %A A010621 njas %I A009393 %S A009393 0,1,3,6,8,5,7,70,72,1311,1309,49214,49216,2653547,2653545,196707438, %T A009393 196707440,19194804703,19194804701,2385684870742,2385684870744, %U A009393 367985503366779,367985503366777,68980888889771126,68980888889771128 %V A009393 0,1,-3,6,-8,5,-7,70,-72,-1311,1309,49214,-49216,-2653547,2653545,196707438, %W A009393 -196707440,-19194804703,19194804701,2385684870742,-2385684870744, %X A009393 -367985503366779,367985503366777,68980888889771126,-68980888889771128 %N A009393 Expansion of ln(1+tanh(x))/exp(x). %t A009393 Log[ 1+Tanh[ x ] ]/Exp[ x ] %K A009393 sign,done,easy %O A009393 0,3 %A A009393 rhh@research.bell-labs.com %E A009393 Extended with signs 03/97 by Olivier Gerard. %I A011261 %S A011261 1,3,6,8,7,3,8,1,0,6,6,4,2,2,0,1,6,7,4,8,4,2,3,6,7,7,8,8,6,6,4,0,2, %T A011261 9,6,5,3,0,4,9,7,8,6,9,7,9,8,1,9,0,8,2,5,9,9,3,4,9,9,2,0,2,4,3,0,2, %U A011261 7,7,4,4,9,2,9,5,9,9,3,5,1,6,0,4,8,5,9,7,5,8,8,5,5,5,6,0,3,0,0,0,9 %N A011261 Decimal expansion of 7th root of 9. %K A011261 nonn,cons %O A011261 1,2 %A A011261 njas %I A036265 %S A036265 3,6,8,8,8,4,6,14,14,6,4,4,2,2,12,14,6,6,12,10,4,6,8,2,8,8,4,28, %T A036265 32,18,18,28,28,16,2,6,6,2,16,28,26,14,16,22,2,14,2,4,6,18,24,16, %U A036265 4,4,12,14,6,10,14,10,22,4,4,26,30,24,22,10,0,4,6,4,6,4,6,14,10 %V A036265 3,-6,8,-8,8,-4,-6,14,-14,6,4,-4,-2,-2,12,-14,6,6,-12,10,-4,-6,8,2,-8,8,4,-28, %W A036265 32,-18,18,-28,28,-16,2,6,-6,-2,16,-28,26,-14,16,-22,2,14,-2,-4,-6,18,-24,16, %X A036265 -4,-4,12,-14,6,10,-14,-10,22,-4,4,-26,30,-24,22,-10,0,-4,6,-4,6,-4,-6,14,-10 %N A036265 4th differences of primes. %K A036265 sign,done %O A036265 1,1 %A A036265 njas %I A016663 %S A016663 3,6,8,8,8,7,9,4,5,4,1,1,3,9,3,6,3,0,2,8,5,2,4,5,5,6,9,7,6,0,0,7,1, %T A016663 7,3,4,3,7,5,2,1,0,1,7,5,7,3,4,9,2,8,3,4,8,4,2,7,4,6,8,7,9,1,9,9,5, %U A016663 4,3,5,9,8,5,3,6,1,6,7,4,1,9,1,1,4,4,7,7,2,3,8,5,9,0,8,2,4,3,5,6,7 %N A016663 Decimal expansion of ln(40). %D A016663 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2. %K A016663 nonn,cons %O A016663 1,1 %A A016663 njas %I A023993 %S A023993 0,3,6,8,8,11,11,14,14,14,17,21,18,20,22,23,21,24,24,27,26,29,29,34,29,30, %T A023993 35,33,32,34,33,39,34,34,37,39,35,38,41,43,40,45,46,50,51,48,49,54,49,50, %U A023993 51,52,49,56,53,55,54,53,58,62,57,58,61,63,56,58,58,61,61,63,62,68,61,65,70 %N A023993 Sum of exponents of primes in multinomial coefficient M(3n;n+2,n-1,n-1). %K A023993 nonn %O A023993 1,2 %A A023993 Clark Kimberling (ck6@cedar.evansville.edu) %I A004715 %S A004715 3,6,8,9,10,13,21,23,27,28,29,31,32,35,36,37,39,40,41,42, %T A004715 43,46,47,52,57,58,61,62,65,67,69,70,72,76,80,81,82,83, %U A004715 85,86,88,89,91,92,94,96,97,98,102,103,105,107,111,112 %N A004715 Positions of ones in binary expansion of arctan(1/2)/Pi. %D A004715 J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, Canad. J. Math., 47 (1995), 262-273. %H A004715 S. Plouffe, The Computation of Certain Numbers Using a Ruler and Compass, J. Integer Seqs. Vol. 1 (1998), #98.1.3. %H A004715 E. W. Weisstein, Link to a section of The World of Mathematics. %H A004715 Steven Finch, Plouffe's Constant %Y A004715 Cf. A028999. %K A004715 nonn,base %O A004715 1,1 %A A004715 sp %I A036558 %S A036558 0,3,6,8,9,11,12,15,17,18,19,20,21,22,24,25,27,29,30,31,32,33,34,35, %T A036558 36,37,39,41,43,44,45,46,47,48,49,50,51,53,54,55,56,57,59,60,61,62, %U A036558 63,65,66,67,68,69,70,71,72,73,75,76,77,78,79,80,81,82,83,84,85,86 %N A036558 Squared distances that can arise in any Barlow packing of spheres of diameter root3 (the scale being the simplest one that makes this an integer sequence). %F A036558 The set of all values of N = xx + xy + yy + 2zz for which x,y,z are integers subject to the proviso that N must be a multiple of 3 if |z| < 2. %Y A036558 Complement of A036559. %K A036558 nonn %O A036558 0,2 %A A036558 J. H. Conway (conway@math.princeton.edu), njas %I A005870 M2516 %S A005870 0,3,6,8,9,11,12,15,17,18,19,20,21,22,25,27,29,30,31,32,33,34,35,36,37, %T A005870 39,41,43,44,45,46,47,48,49,51,53,54,55,56,57,59,61,63,65,66,67,68,69,70,71,72,75 %N A005870 Numbers represented by hexagonal close-packing. %D A005870 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114. %K A005870 easy,nonn %O A005870 0,2 %A A005870 njas %I A005652 M2517 %S A005652 1,3,6,8,9,11,14,16,17,19,21,22,24,27,29,30,32,35,37,40,42,43,45,48,50, %T A005652 51,53,55,56,58,61,63,64,66,69,71,74,76,77,79,82,84,85,87,90,92,95,97,98,100,103,105 %N A005652 Sum of 2 terms is never a Fibonacci number. %C A005652 The set of all n such that the integer multiple of (1+sqrt(5))/2 nearest n is greater than n. Chow-Long paper proves this, establishes a connection with continued fractions, gives generalizations and other references on this sequence and related sequences - comment from Timothy Y. Chow (tchow@alum.mit.edu) %D A005652 K. Alladi et al., On additive partitions of integers, Discrete Math., 22 (1978), 201-211. %H A005652 T. Y. Chow and C. D. Long, Additive partitions and continued fractions, Ramanujan J., 3 (1999), 55-72 [set alpha=(1+sqrt(5))/2 in Theorem 2 to get A005652 and A005653]. %Y A005652 Complement of A005653. %K A005652 nonn,easy,nice,more %O A005652 1,2 %A A005652 sp,njas %I A047401 %S A047401 0,1,3,6,8,9,11,14,16,17,19,22,24,25,27,30,32,33,35,38,40,41,43,46,48, %T A047401 49,51,54,56,57,59,62,64,65,67,70,72,73,75,78,80,81,83,86,88,89,91,94, %U A047401 96,97,99,102,104,105,107 %N A047401 Congruent to {0, 1, 3, 6} mod 8. %K A047401 nonn %O A047401 0,3 %A A047401 njas %I A024707 %S A024707 3,6,8,9,11,15,16,18,20,22,24,27,30,32,33,34,39,40,41,44,45,48,50,51,52,55, %T A024707 56,58,62,65,68,70,73,75,77,78,79,80,81,83,85,87,90,92,93,95,96,98,102,103, %U A024707 107,108,112,113,114,118,119,123,124,125,126,129,131,133,134,138,139,140 %N A024707 Positions of multiplies of 5 in A024702. %K A024707 nonn %O A024707 1,1 %A A024707 Clark Kimberling (ck6@cedar.evansville.edu) %I A032489 %S A032489 3,6,8,9,12,13,14,19,22,32,35,37,80,86,102,118,135,159,205,269,430,435, %T A032489 535,547,830,1020,1030,1079,1089,1112,1189,2244,2397,4497,5888,9549, %U A032489 15942,16855,19736,21936,26744,26926,36744,51142,53155,53532 %N A032489 225*2^n+1 is prime. %H A032489 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A032489 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A032489 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %H A032489 R. Ballinger and W. Keller, More information %K A032489 nonn,hard %O A032489 0,1 %A A032489 njas %I A005622 M2518 %S A005622 1,3,6,8,9,17,25,28,79,119,132,281,437 %N A005622 Spiral sieve using Fibonacci numbers. %D A005622 H. W. Gould, Some sequences generated by spiral sieving methods, Fib. Quart., 12 (1974), 393-397. %H A005622 Index entries for sequences generated by sieves %K A005622 nonn %O A005622 1,2 %A A005622 njas %I A055073 %S A055073 3,6,8,10,11,12,16,17,18,30,32,37,40,41,43,44,45,55,57,68,74,75,76,84, %T A055073 85,94,95,101,106,113,128,131,136,138,149,154,159,171,172,178,179,180, %U A055073 181,182,183,184,212,214,226,228,229,240,241,245,258,259,260,275,278 %N A055073 Pointers to base 3 digits where primes occur in A055072. %e A055073 012101112202122100101102110111112120121122200201... %e A055073 --|..... sum 0+1+2 = prime 3 at position 3 %e A055073 -----|.. sum 0+1+2+1+0+1 = prime 5 at position 6 %e A055073 -------| sum 0+1+2+1+0+1+1+1 = prime 7 at position 8 %K A055073 nonn,base %O A055073 0,1 %A A055073 Patrick De Geest (pdg@worldofnumbers.com), Apr 2000. %I A012132 %S A012132 3,6,8,10,11,13,15,16,18,20,21,23,26,27,28,31,33,36,37,38,40,41,43,44, %T A012132 45,46,48,49,51,52,53,54,55,56,57,58,59,61,62,63,64,66,67,68,71,73,74, %U A012132 75,76,77,78,80,81,83,86,88,89,91,92,93 %N A012132 Numbers z such x(x+1)+y(y+1)=z(z+1) is solvable. %D A012132 H. Finner and K. Strassburger, Increasing sample sizes do not necessarily increase the power of UMPU-tests for 2x2-tables. Metrika, 54, 77-91, (2001). %D A012132 Aviezri S. Fraenkel, Diophantine equations involving generalized triangular and tetrahedral numbers, pp. 99-114 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971. %D A012132 H. Harborth, Fermat-like binomial equations, Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose/Ca., August 1986, 1-5 (1988). %Y A012132 Complement of A027861 - Michael Somos, June 5, 2000. %K A012132 nonn %O A012132 1,1 %A A012132 sander@win.tue.nl (Sander van Rijnswou) %E A012132 More terms and references from Klaus Strassburger (strass@dfi.uni-duesseldorf.de), Feb 09 2000 %I A023983 %S A023983 3,6,8,10,12,14,19,17,18,21,24,24,29,27,29,27,31,29,33,35,37,39,41,40,40,45, %T A023983 45,43,48,48,54,46,48,51,52,51,56,57,59,58,57,60,61,62,65,64,69,63,68,66,69, %U A023983 74,76,74,78,75,76,81,82,81,85,87,83,77,81,82,84,81,87,86,93,87,91,92,93,96 %N A023983 Sum of exponents in prime-power factorization of multinomial coefficient M(5n;2n,2n,n). %K A023983 nonn %O A023983 1,1 %A A023983 Clark Kimberling (ck6@cedar.evansville.edu) %I A047282 %S A047282 1,3,6,8,10,13,15,17,20,22,24,27,29,31,34,36,38,41,43,45,48,50,52,55, %T A047282 57,59,62,64,66,69,71,73,76,78,80,83,85,87,90,92,94,97,99,101,104,106, %U A047282 108,111,113,115,118,120,122 %N A047282 Congruent to {1, 3, 6} mod 7. %K A047282 nonn %O A047282 0,2 %A A047282 njas %I A001066 %S A001066 3,6,8,10,14,15,16,20,21,24,28,30,35,36,42,45,48,48,52, %T A001066 55,56,63,66,70,72,78,80,90,91,96,99,104,105,110,120,126, %U A001066 132,133,136,143,153,156,160,168,171,182,190,195,198,210 %N A001066 Dimensions of real simple Lie algebras. %D A001066 N. Jacobson, Lie Algebras. Wiley, NY, 1962, p. 146. %D A001066 Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652. %Y A001066 Cf. A003038. %K A001066 nonn,nice,easy %O A001066 1,1 %A A001066 reb@math.berkeley.edu (Richard E. Borcherds) %I A043549 %S A043549 1,1,1,3,6,8,10,15,21,28,32,41,45,55,66,78,91,98,113,128,136,153,171, %T A043549 190,210,231,242,265,288,313,325,351,378,406,435,465,496,512,545,578, %U A043549 613,648,666,703,741,780,820,861,903,946,968 %N A043549 Least k for which the integers Floor(2k/m) for m=2,3,...,n are distinct. %K A043549 nonn %O A043549 1,4 %A A043549 Clark Kimberling, ck6@cedar.evansville.edu %I A026604 %S A026604 1,3,6,8,11,12,15,16,18,20,23,24,27,28,30,31,33,36,39,40,42,43, %T A026604 45,48,50,53,54,56,59,60,63,64,66,67,69,72,75,76,78,79,81,84, %U A026604 86,89,90,91,93,96,98,101,102,105,106,108,111,112,114,115,117 %N A026604 a(n) = s(1) + s(2) + ... + s(n), where s = A026600. %K A026604 nonn %O A026604 1,2 %A A026604 Clark Kimberling, ck6@cedar.evansville.edu %I A013642 %S A013642 3,6,8,11,12,15,18,20,24,27,30,35,38,39,40,42,48,51,56,63,66,68,72,80,83, %T A013642 84,87,90,99,102,104,105,110,120,123,132,143,146,147,148,150,152,156,168, %U A013642 171,182,195,198,200,203,210,224,227,228,230,231,235,240,255,258,260,264 %N A013642 Continued fraction for sqrt(n) has period 2. %D A013642 Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!) %K A013642 nonn %O A013642 1,1 %A A013642 Clark Kimberling (ck6@cedar.evansville.edu) %I A047219 %S A047219 1,3,6,8,11,13,16,18,21,23,26,28,31,33,36,38,41,43,46,48,51,53,56,58, %T A047219 61,63,66,68,71,73,76,78,81,83,86,88,91,93,96,98,101,103,106,108,111, %U A047219 113,116,118,121,123,126,128 %N A047219 Congruent to {1, 3} mod 5. %F A047219 a(n)=Floor[(5n-3)/2] - Santi Spadaro (spados@katamail.com), Jul 24 2001 %Y A047219 Cf. A000566. %K A047219 nonn %O A047219 0,2 %A A047219 njas %I A004957 %S A004957 0,3,6,8,11,14,16,19,21,24,27,29,32,35,37,40,42,45,48,50, %T A004957 53,55,58,61,63,66,69,71,74,76,79,82,84,87,90,92,95,97, %U A004957 100,103,105,108,110,113,116,118,121,124,126,129,131,134 %N A004957 Ceiling of n*tau^2. %Y A004957 Essentially same as A026352. %K A004957 nonn %O A004957 0,2 %A A004957 njas %I A026352 %S A026352 1,3,6,8,11,14,16,19,21,24,27,29,32,35,37,40,42,45,48,50,53,55, %T A026352 58,61,63,66,69,71,74,76,79,82,84,87,90,92,95,97,100,103,105, %U A026352 108,110,113,116,118,121,124,126,129,131,134,137,139,142,144 %N A026352 [ n*tau ]+n+1. %C A026352 a(n) = greatest k such that s(k) = n+1, where s = A026350. %Y A026352 Essentially same as A004957. %K A026352 nonn %O A026352 0,2 %A A026352 Clark Kimberling, ck6@cedar.evansville.edu %I A047399 %S A047399 0,3,6,8,11,14,16,19,22,24,27,30,32,35,38,40,43,46,48,51,54,56,59,62, %T A047399 64,67,70,72,75,78,80,83,86,88,91,94,96,99,102,104,107,110,112,115,118, %U A047399 120,123,126,128,131,134,136 %N A047399 Congruent to {0, 3, 6} mod 8. %K A047399 nonn %O A047399 0,2 %A A047399 njas %I A057349 %S A057349 3,6,8,11,14,17,19,22,25,27,30,33,36,38,41,44,46,49,52,55,57,60,63,65, %T A057349 68,71,74,76,79,82,84,87,90,93,95,98,101,103,106,109,112,114,117,120, %U A057349 122,125,128,131,133,136,139,141,144,147,150,152,155,158,160,163,166 %N A057349 Leap years in the Hebrew Calendar starting in year 1 (3761 BCE). The leap year has an extra -month-. %C A057349 A Hebrew year approximates a solar year with 12 and 7/19 lunar months (or 19 years with 235 months, the 19-year Metonic cycle) %D A057349 Dershowitz and Reingold, Calendrical Calculations, Cambridge University Press, 1997. %H A057349 Calendrical Calculations %H A057349 Calendar Applet %F A057349 a(n) = floor((19n + 5)/7), ((1 + 7*n) mod 19) < 7 %Y A057349 A008685, Hebrew month pattern A057350, A057347. %K A057349 nonn,easy %O A057349 1,1 %A A057349 Mitch Harris (maharri@cs.uiuc.edu) %I A022851 %S A022851 0,3,6,8,11,14,17,20,23,25,28,31,34,37,40,42,45,48,51,54,57,59,62,65, %T A022851 68,71,74,76,79,82,85,88,91,93,96,99,102,105,107,110,113,116,119, %U A022851 122,124,127,130,133,136,139,141,144,147,150,153,156,158,161,164 %N A022851 a(n) = integer nearest nx, where x = sqrt(8). %K A022851 nonn %O A022851 0,2 %A A022851 Clark Kimberling (ck6@cedar.evansville.edu) %I A050503 %S A050503 0,1,3,6,8,11,14,17,20,23,26,30,33,37,41,44,48,52,56,60,64,68,72, %T A050503 76,80,85,89,93,98,102,106,111,115,120,124,129,134,138,143,148, %U A050503 152,157,162,167,171,176,181,186,191,196,201,205,210,215,220,225 %N A050503 Nearest integer to n*ln(n). %C A050503 The prime number theorem states that the n-th prime is asymptotic to n*ln(n). %D A050503 Cf. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 8. %Y A050503 Cf. A000040, A050502, A050504. %K A050503 nonn %O A050503 1,3 %A A050503 njas, Dec 27 1999 %I A036434 %S A036434 1,3,6,8,11,16,17,20,22,23,27,29,35,36,40,41,44,46,47,53,54,57,60,65, %T A036434 67,68,70,76,77,79,80,83,87,88,92,93,94,100,101,106,107,114,116,117, %U A036434 121,125,128,131,132,134,135,140,142,148,155,156,157,158,161,164,166 %N A036434 Integers which cannot be written as k+tau(k) for some k. %C A036434 Invented by the HR concept formation program. %H A036434 S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2. %H A036434 S. Colton, HR - Automatic Theory Formation in Pure Mathematics %e A036434 None of 1,2,3,4,5,6,7 are such that k+tau(k)=8, so 8 is in the sequence %Y A036434 Cf. A036438. %K A036434 nonn %O A036434 1,2 %A A036434 Simon Colton (simonco@cs.york.ac.uk) %I A006048 M2519 %S A006048 1,3,6,8,12,18,21,27,36,38,42,48,52,60,72,78,90,108,111,117,126,132,144, %T A006048 162,171,189,216,218,222,228,232,240,252,258,270,288,292,300,312,320,336,360,372,396 %N A006048 Entries in first n rows of Pascal's triangle not divisible by 3. %D A006048 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197. %D A006048 A. Lakhtakia et al., Fractal sequences derived from the self-similar extensions of the Sierpinski gasket, J. Phys. A 21 (1988), 1925-1928. %D A006048 A. Lakhtakia and R. Messier, Self-similar sequences and chaos from Gauss sums, Computers and Graphics, 13 (1989), 59-62. %K A006048 nonn %O A006048 0,2 %A A006048 jos %I A049827 %S A049827 0,1,3,6,8,13,13,20,22,25,27,38,32,45,45,46,50,63,59,74,64,71,77,96 %N A049827 a(n)=T(2n-1,n)+T(2n,n+1)+...+T(3n-3,2n-2)=sum over a period of n-th diagonal of array T given by A049816. %K A049827 nonn %O A049827 1,3 %A A049827 Clark Kimberling, ck6@cedar.evansville.edu %I A024608 %S A024608 3,6,8,13,15,19,21,25,29,32,36,38,41,44,49,51,53,58,63,65,68,74,79,81,82,84, %T A024608 92,93,95,99,100,105,112,114,119,122,125,129,131,134,135,139,144,147,150 %N A024608 Positions of even numbers in A003136. %K A024608 nonn %O A024608 1,1 %A A024608 Clark Kimberling (ck6@cedar.evansville.edu) %I A046669 %S A046669 1,3,6,8,13,15,22,24,27,29,40,42,55,57,60,62,79,81,100,102,105, %T A046669 107,130,132,137,139,142,144,173,175,206,208,211,213,218,220,257, %U A046669 259,262,264,305,307,350,352,355,357,404,406,413,415,418,420,473 %N A046669 Partial sums of A020639. %D A046669 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1. %K A046669 nonn,nice,easy %O A046669 1,2 %A A046669 njas %I A046670 %S A046670 1,3,6,8,13,16,23,25,28,33,44,47,60,67,72,74,91,94,113,118,125,136,159, %T A046670 162,167,180,183,190,219,224,255,257,268,285,292,295,332,351,364,369, %U A046670 410,417,460,471,476,499,546,549,556,561,578,591,644,647,658,665,684 %N A046670 Partial sums of A006530. %D A046670 Handbook of Number Theory, D. S. Mitrinovic et al., Kluwer, Section IV.1. %K A046670 nonn,nice,easy %O A046670 0,2 %A A046670 njas %E A046670 More terms from James A. Sellers (sellersj@math.psu.edu) %I A051212 %S A051212 0,3,6,8,15,21,24,26,35,39,44,48,54,63,69,71,80,89,90,96,99,111,120, %T A051212 125,134,143,156,159,168,186,189,195,201,215,224,225,246,255,261,279, %U A051212 288,296,300,314,323,341,351,360,369,384,390,399,404,429,431,440,444 %N A051212 Of the form x^2-10^y >= 0. %K A051212 nonn %O A051212 0,2 %A A051212 dww %I A039567 %S A039567 1,3,6,8,16,18,31,33,41,43,54,70,81,83,91,93,102,110,139,147,156,158, %T A039567 166,168,179,195,206,208,216,218,227,235,259,269,271,273,279,295,329, %U A039567 345,351,353,355,365,389,397,406,408,416,418,429,445,456,458,466,468 %N A039567 Representation in base 5 has same number of 0's, 2's and 4's. %K A039567 nonn,base,easy %O A039567 0,2 %A A039567 Olivier Gerard (ogerard@ext.jussieu.fr) %I A032912 %S A032912 1,3,6,8,16,18,31,33,41,43,81,83,91,93,156,158,166,168,206,208, %T A032912 216,218,406,408,416,418,456,458,466,468,781,783,791,793,831, %U A032912 833,841,843,1031,1033,1041,1043,1081,1083,1091,1093,2031,2033 %N A032912 n-th number whose set of base 5 digits is {1,3}. %K A032912 nonn,base %O A032912 1,2 %A A032912 Clark Kimberling, ck6@cedar.evansville.edu %I A033031 %S A033031 1,3,6,8,16,18,31,33,41,43,81,83,91,93,156,158,166,168,206,208, %T A033031 216,218,406,408,416,418,456,458,466,468,781,783,791,793,831, %U A033031 833,841,843,1031,1033,1041,1043,1081,1083,1091,1093,2031,2033 %N A033031 n-th number all of whose base 5 digits are odd. %K A033031 nonn,base %O A033031 1,2 %A A033031 Clark Kimberling, ck6@cedar.evansville.edu %I A038064 %S A038064 3,6,8,18,48,124,312,810,2184,5928,16104,44220,122640,341796, %T A038064 956576,2690010,7596480,21524412,61171656,174336264,498111952, %U A038064 1426419852,4093181688,11767874940,33891544368,97764131640 %V A038064 3,-6,8,-18,48,-124,312,-810,2184,-5928,16104,-44220,122640,-341796, %W A038064 956576,-2690010,7596480,-21524412,61171656,-174336264,498111952, %X A038064 -1426419852,4093181688,-11767874940,33891544368,-97764131640 %N A038064 prod{k=1..inf}(1/(1-x^k)^a(k)) = 1+3x. %H A038064 N. J. A. Sloane, Euler transform %Y A038064 Cf. A038063-A038070. %K A038064 sign,done %O A038064 1,1 %A A038064 Christian G. Bower (bowerc@usa.net), Jan 1999. %I A059361 %S A059361 1,3,6,8,81,96,8192,36669429,36675584 %N A059361 a(1)=1; a(n) = A059333(a(n-1)) + a(n-1). %Y A059361 Cf. A059333. %K A059361 more,nonn %O A059361 1,2 %A A059361 Naohiro Nomoto (6284968128@geocities.co.jp), Jan 27 2001 %I A021275 %S A021275 0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3, %T A021275 6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0, %U A021275 0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6,9,0,0,3,6 %N A021275 Decimal expansion of 1/271. %K A021275 nonn,cons %O A021275 0,3 %A A021275 njas %I A059626 %S A059626 0,3,6,9,2,5,8,1,4,7,30,33,36,39,32,35,38,31,34,37,60,63,66,69,62,65, %T A059626 68,61,64,67,90,93,96,99,92,95,98,91,94,97,20,23,26,29,22,25,28,21,24, %U A059626 27,50,53,56,59,52,55,58,51,54,57,80,83,86,89,82,85,88,81,84,87,10,13 %N A059626 Generalised nim sum n + n + n in base 10; carryless multiplication 3 X n base 10. %H A059626 Index entries for sequences that are permutations of the natural numbers %Y A059626 Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A008592 for carryless 10 X n. %K A059626 nonn %O A059626 0,2 %A A059626 Henry Bottomley (se16@btinternet.com), Feb 19 2001 %I A013663 %S A013663 1,0,3,6,9,2,7,7,5,5,1,4,3,3,6,9,9,2,6,3,3,1,3,6,5,4,8,6,4,5,7,0,3, %T A013663 4,1,6,8,0,5,7,0,8,0,9,1,9,5,0,1,9,1,2,8,1,1,9,7,4,1,9,2,6,7,7,9,0, %U A013663 3,8,0,3,5,8,9,7,8,6,2,8,1,4,8,4,5,6,0,0,4,3,1,0,6,5,5,7,1,3,3,3,3 %N A013663 Decimal expansion of zeta(5). %D A013663 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811. %H A013663 Robert J. Harley, Zeta(3), Zeta(5), .., Zeta(99) 10000 digits (txt, 400 KB) %H A013663 Simon Plouffe, Other interesting computations %Y A013663 Cf. A002117, A013667, A013669, A013671, A013675, A013677. %K A013663 cons,nonn %O A013663 1,3 %A A013663 njas %I A020850 %S A020850 1,0,3,6,9,5,1,6,9,4,7,3,0,4,2,5,2,6,8,3,6,1,2,9,3,6,6,1,0,1,4,2,6, %T A020850 4,8,5,8,4,8,4,6,1,3,3,0,0,3,2,9,3,8,1,9,7,9,2,7,9,5,8,8,9,1,2,6,5, %U A020850 9,2,7,8,1,0,9,5,3,4,2,6,3,4,1,1,0,6,4,5,2,2,0,1,1,7,5,2,0,0,0,9,2 %N A020850 Decimal expansion of 1/sqrt(93). %K A020850 nonn,cons %O A020850 0,3 %A A020850 njas %I A019700 %S A019700 3,6,9,5,9,9,1,3,5,7,1,6,4,4,6,2,6,3,3,4,8,5,4,6,2,8,0,3,8,5,8,2,3, %T A019700 8,6,8,7,2,9,0,7,8,7,5,2,8,6,7,6,5,9,5,0,8,3,4,9,9,9,3,4,8,1,4,4,7, %U A019700 9,7,8,4,0,0,7,3,9,5,5,3,9,9,9,8,3,8,5,9,2,3,3,2,3,9,3,1,9,0,2,4,3 %N A019700 Decimal expansion of 2*Pi/17. %K A019700 nonn,cons %O A019700 0,1 %A A019700 njas %I A049341 %S A049341 3,6,9,6,6,3,9,3,3,6,9,6,6,3,9,3,3,6,9,6,6,3,9,3,3,6,9,6,6,3,9,3,3,6,9, %T A049341 6,6,3,9,3,3,6,9,6,6,3,9,3,3,6,9,6,6,3,9,3,3,6,9,6,6,3,9,3,3,6,9,6,6,3, %U A049341 9,3,3,6,9,6,6,3,9,3,3,6,9,6,6,3,9,3,3,6,9,6,6,3,9,3,3,6,9,6,6,3,9,3,3,6,9,6,6,3,9,3 %N A049341 a(n+1) = iterated sum of digits of a(n) + a(n-1). %F A049341 Period 8. %e A049341 After 6,9 we get 6+9 = 15 -> 1+5 = 6. %Y A049341 Cf. A030132, A049342. %K A049341 base,nonn %O A049341 0,1 %A A049341 Damir Olejar (damir666@hotmail.com) %I A021077 %S A021077 0,1,3,6,9,8,6,3,0,1,3,6,9,8,6,3,0,1,3,6,9,8,6,3,0,1,3,6,9,8,6,3,0, %T A021077 1,3,6,9,8,6,3,0,1,3,6,9,8,6,3,0,1,3,6,9,8,6,3,0,1,3,6,9,8,6,3,0,1, %U A021077 3,6,9,8,6,3,0,1,3,6,9,8,6,3,0,1,3,6,9,8,6,3,0,1,3,6,9,8,6,3,0,1,3 %N A021077 Decimal expansion of 1/73. %K A021077 nonn,cons %O A021077 0,3 %A A021077 njas %I A000748 M2520 N0995 %S A000748 1,3,6,9,9,0,27,81,162,243,243,0,729,2187,4374,6561,6561, %T A000748 0,19683,59049,118098,177147,177147,0,531441,1594323,3188646, %U A000748 4782969,4782969,0 %V A000748 1,-3,6,-9,9,0,-27,81,-162,243,-243,0,729,-2187,4374,-6561,6561, %W A000748 0,-19683,59049,-118098,177147,-177147,0,531441,-1594323,3188646, %X A000748 -4782969,4782969,0 %N A000748 Expansion of bracket function. %D A000748 H. W. Gould, Binomial coefficients, the bracket function, and compositions with relatively prime summands, Fib. Quart. 2 (1964), 241-260. %F A000748 G.f.: 1/((1+x)^3-x^3). %Y A000748 Cf. A000749, A000750, A001659. %K A000748 sign,done,easy %O A000748 0,2 %A A000748 njas %I A057083 %S A057083 1,3,6,9,9,0,27,81,162,243,243,0,729,2187,4374,6561,6561,0,19683,59049, %T A057083 118098,177147,177147,0,531441,1594323,3188646,4782969,4782969,0, %U A057083 14348907,43046721,86093442,129140163,129140163,0 %V A057083 1,3,6,9,9,0,-27,-81,-162,-243,-243,0,729,2187,4374,6561,6561,0, %W A057083 -19683,-59049,-118098,-177147,-177147,0,531441,1594323,3188646, %X A057083 4782969,4782969,0,-14348907,-43046721,-86093442,-129140163,-129140163,0 %N A057083 Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/(1-3*x+3*x^2). %C A057083 With different sign pattern, see A000748. %D A057083 A. F. Horadam, Special properties of the sequence W_n(a,b;p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=3, q=-3. %D A057083 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (38) and (45),lhs, m=3. %H A057083 Index entries for sequences related to Chebyshev polynomials. %F A057083 a(n)=S(n,sqrt(3))*(sqrt(3))^n with S(n,x):=U(n,x/2), Chebyshev polynomials of 2nd kind, A049310. %F A057083 a(2*n)= A057078(n)*3^n; a(2*n+1)= A010892(n)*3^(n+1). %F A057083 G.f.: 1/(1-3*x+3*x^2). %Y A057083 A049310, A057078, A010892, A000748. %K A057083 easy,sign,done %O A057083 0,2 %A A057083 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Aug 11 2000 %I A011383 %S A011383 1,3,6,9,9,8,7,3,1,7,7,3,9,7,4,5,7,0,2,1,0,9,1,7,1,6,9,0,6,5,1,9,8, %T A011383 2,1,8,3,1,9,8,6,3,0,3,7,4,6,9,3,9,5,9,9,0,4,8,4,2,2,2,9,0,4,0,7,2, %U A011383 3,6,2,5,4,8,7,1,1,5,4,3,2,8,7,7,6,5,7,8,0,9,4,0,2,1,2,3,3,5,1,8,9 %N A011383 Decimal expansion of 9th root of 17. %K A011383 nonn,cons %O A011383 1,2 %A A011383 njas %I A007844 %S A007844 1,3,6,9,9,12,15,18,18,21,24,27,27,27,30,33,36,36,39,42,45,45,48,51,54,54, %T A007844 54,57,60,63,63,66,69,72,72,75,78,81,81,81,81,84,87,90,90,93,96,99,99, %U A007844 102,105,108,108,108,111,114,117,117,120,123,126,126,129,132,135,135,135 %N A007844 Least positive integer k for which 3^n divides k!. %D A007844 F. Smarandache, "Only Problems, not Solutions!", Xiquan Publ., Phoenix-Chicago, 1993 %D A007844 H. Ibstedt, Smarandache Primitive Numbers, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 216-229. %H A007844 M. L. Perez et al., eds., Smarandache Notions Journal %Y A007844 Cf. A007843, A007845. %K A007844 nonn %O A007844 0,2 %A A007844 Bruce Dearden and Jerry Metzger [ metzger@rs1.cc.und.nodak.edu ], R. Muller %I A057338 %S A057338 1,3,6,9,9,15,15,21,24,24,24,36,36,36,48,51,51,57,57,75,75,75,75,90,90, %T A057338 90,90,90,90,114,114,114,114,114,126,138,138,138,138,156,156,180,180, %U A057338 180,198,198,198,207,207,207,207,207,207,207,207,237,237,237,237,267 %N A057338 Occurrences of most frequently occurring number in 1-to-n multiplication cube. %e A057338 M(n) is the array in which m(x,y,z)=x*y*z for x = 1 to n, y = 1 to n, and z = 1 to n. In M(7) the most frequently occurring numbers are 12 and 24. They occur 15 times, so a(7) = 15. %Y A057338 Cf. A057142, A057339, A057340, A057341, A057344. %K A057338 nonn %O A057338 1,2 %A A057338 Neil Fernandez (primeness@borve.demon.co.uk), Aug 28 2000 %E A057338 More terms from David Wilson (wilson@aprisma.com), Aug 28 2001 %I A022304 %S A022304 1,3,6,9,10,12,13,16,18,19,21,24,25,27,28,30,33,36,37,39,42,44,45, %T A022304 47,48,51,54,55,57,60,63,64,66,67,69,72,74,75,78,81,82,84,87,90,91, %U A022304 93,94,97,99,100,102,105,108,109,111,112,114,117,118,120,121,124 %N A022304 Index of n-th 1 in A022303. %K A022304 nonn %O A022304 0,2 %A A022304 Clark Kimberling (ck6@cedar.evansville.edu) %I A055632 %S A055632 3,6,9,10,12,14,18,20,22,24,26,27,28,30,34,36,38,40,44,46,48,50,52,54, %T A055632 56,58,60,62,66,68,70,72,74,76,80,81,82,86,88,90,92,94,96,98,100,102, %U A055632 104,106,108,112,116,118,120,122,124,130,132,134,136,140,142,144,146 %N A055632 Sum of totient function of primes dividing n is a prime. %e A055632 If n=(2^a)*(3^b)*(5^c)*(7^d)*(11^e), then prime-factor set is {2,3,5,7,11}. The totient function values of this set are {1,2,4,6,10} and the sum is 1+2+4+6+10=23. Observe that this sequence includes even numbers and for all p primes as (a phi-sum) an infinite number of solutions exist, like e.g. (2^w)*p, with 1+p-1=p Phi-sum over its factors. %Y A055632 A001221, A006093, A053571, A055631. %K A055632 nonn %O A055632 1,1 %A A055632 Labos E. (labos@ana1.sote.hu), Jun 06 2000 %I A055264 %S A055264 0,1,3,6,9,10,12,15,18,19,21,24,27,28,30,33,36,37,39,42,45,46,48,51,54, %T A055264 55,57,60,63,64,66,69,72,73,75,78,81,82,84,87,90,91,93,96,99,100,102, %U A055264 105,108,109,111,114,117,118,120,123,126,127,129,132,135,136,138,141 %N A055264 Possible values of A055263; numbers equal to 0, 1, 3 or 6 modulo 9. %F A055264 a(n) =a(n-4)+9 =9*floor[n/4]+[n mod 4]*(1+[n mod 4])/2 %Y A055264 Cf. A055263. %K A055264 easy,nonn %O A055264 0,3 %A A055264 Henry Bottomley (se16@btinternet.com), May 08 2000 %I A061904 %S A061904 1,3,6,9,10,12,15,18,21,30,39,45,48,51,60,90,100,102,105,111,120,150, %T A061904 180,201,210,249,300,318,321,348,351,390,450,480,501,510,549,600,900, %U A061904 1000,1002,1005,1011,1020,1050,1101,1110,1149,1200,1500,1761,1800,2001 %N A061904 The iterative cycle: n -> sum of digits of n^2 has only one distinct element. %C A061904 Since the only numbers invariant under this iteration are 1 and 9, n is contained in this sequence iff the sum of digits of n^2 is 1 or 9. %e A061904 6 -> 3+6 = 9 -> 8+1 = 9 thus 9 is the only element of the iterative cycle of 6. 12 -> 1+4+4 = 9 -> 8+1 = 9 ... %Y A061904 Cf. A007953, A004159, A061903 - A061910. %K A061904 nonn,base %O A061904 1,2 %A A061904 Asher Auel (asher.auel@reed.edu), 17 May 2001 %I A024795 %S A024795 3,6,9,11,12,14,17,18,19,21,22,24,26,27,27,29,30,33,33,34,35,36,38,38,41,41, %T A024795 42,43,44,45,46,48,49,50,51,51,53,54,54,54,56,57,57,59,59,61,62,62,65,66,66, %U A024795 66,67,68,69,69,70,72,73,74,74,75,75,76,77,77,78,81,81,81,82,83,83,84,86,86 %N A024795 Sum of 3 nonzero squares, including repetitions. %H A024795 Index entries for sequences related to sums of squares %Y A024795 Cf. A000408. %K A024795 nonn %O A024795 1,1 %A A024795 Clark Kimberling (ck6@cedar.evansville.edu) %I A000408 %S A000408 3,6,9,11,12,14,17,18,19,21,22,24,26,27,29,30,33,34,35,36,38,41,42,43,44, %T A000408 45,46,48,49,50,51,53,54,56,57,59,61,62,65,66,67,68,69,70,72,73,74,75,76, %U A000408 77,78,81,82,83,84,86,88,89,90,91,93,94,96,97,98,99,101,102,104,105,106 %N A000408 Sum of 3 nonzero squares. %H A000408 Index entries for sequences related to sums of squares %Y A000408 Cf. A024795. %K A000408 nonn,easy %O A000408 1,1 %A A000408 njas,jhc %I A025321 %S A025321 3,6,9,11,12,14,17,18,19,21,22,24,26,29,30,34,35,36,42,43,44,45,46,48,49, %T A025321 50,53,56,61,65,67,68,70,72,73,76,78,82,84,88,91,93,96,97,104,106,109, %U A025321 115,116,120,133,136,140,142,144,145,157,163,168,169,172,176,180,184,190 %N A025321 Sum of 3 nonzero squares in exactly 1 way. %H A025321 Index entries for sequences related to sums of squares %H A025321 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A025321 Cf. A000408. %K A025321 nonn %O A025321 1,1 %A A025321 dww %I A047400 %S A047400 1,3,6,9,11,14,17,19,22,25,27,30,33,35,38,41,43,46,49,51,54,57,59,62, %T A047400 65,67,70,73,75,78,81,83,86,89,91,94,97,99,102,105,107,110,113,115,118, %U A047400 121,123,126,129,131,134,137 %N A047400 Congruent to {1, 3, 6} mod 8. %K A047400 nonn %O A047400 0,2 %A A047400 njas %I A054414 %S A054414 1,3,6,9,11,14,17,19,22,25,28,30,33,36,38,41,44,47,49,52,55,57,60,63, %T A054414 66,68,71,74,76,79,82,84,87,90,93,95,98,101,103,106,109,112,114,117, %U A054414 120,122,125,128,131,133,136,139,141,144,147,150,152,155,158,160,163 %N A054414 If theta = ln(2)/ln(3) then a(n) = 1 + Floor(n/(1-theta)) %C A054414 These numbers appear in connection with the 3x+1 problem. %e A054414 a(5) = 1 + floor(5/(1-theta))= 1 + floor(5/0.3690702464)= 1 + floor(13,54..) = 14 %p A054414 Digits:=500: it:=evalf(ln(2)/ln(3)): for n from 0 to 200 do printf(%d,,1+floor(n/(1-it))) od: %K A054414 easy,nonn %O A054414 0,2 %A A054414 B.Schaaf (m.m.schaaf-visch@wxs.nl), May 20 2000 %E A054414 More terms from James A. Sellers (sellersj@math.psu.edu), May 23 2000 %I A003252 M2521 %S A003252 3,6,9,12,15,18,21,23,26,29,32,35,38,41,44,47,50,53,56,59,61,64, %T A003252 67,70,73,76,79,81,83,86,89,92,95,98,101,104,107,110,113,116,119,121,124 %N A003252 Related to Fibonacci representations. %D A003252 L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386. %K A003252 nonn %O A003252 1,1 %A A003252 njas %I A008486 %S A008486 1,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54, %T A008486 57,60,63,66,69,72,75,78,81,84,87,90,93,96,99,102,105, %U A008486 108,111,114,117 %N A008486 Expansion of (1-x^3 )/(1-x)^3. %F A008486 3n, n >= 1. %K A008486 nonn %O A008486 0,2 %A A008486 njas %I A008585 %S A008585 0,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54, %T A008585 57,60,63,66,69,72,75,78,81,84,87,90,93,96,99,102,105, %U A008585 108,111,114,117,120,123,126,129,132,135,138,141,144 %N A008585 Multiples of 3. %H A008585 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 315 %K A008585 nonn %O A008585 0,2 %A A008585 njas %I A031193 %S A031193 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69, %T A031193 72,75,78,81,84,87,90,93,96,99,102,105,108,111,114,117,120,123,126, %U A031193 129,132,135,138,141,144,147,150,153,156,159,162,165,168,171,174 %N A031193 Numbers having period-22 5-digitized sequences. %H A031193 E. W. Weisstein, Link to a section of The World of Mathematics. %K A031193 nonn %O A031193 0,1 %A A031193 Eric W. Weisstein (eric@weisstein.com) %I A036686 %S A036686 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,50,94 %N A036686 Single-heap P-positions in misere version of Grundy's game. %D A036686 Handbook of Combinatorics, North-Holland '95, p. 1963. %K A036686 nonn,fini %O A036686 0,1 %A A036686 njas %I A059563 %S A059563 3,6,9,12,15,18,21,24,27,30,33,37,40,43,46,49,52,55,58,61,64,67,70,74, %T A059563 77,80,83,86,89,92,95,98,101,104,108,111,114,117,120,123,126,129,132, %U A059563 135,138,141,145,148,151,154,157,160,163,166,169,172,175,178,182,185 %N A059563 Beatty sequence for e+1/e. %D A059563 Fraenkel, Aviezri S.; Levitt, Jonathan; Shimshoni, Michael; Characterization of the set of values f(n)=[n alpha], n=1,2,... Discrete Math.2 (1972),no.4,335-345. %H A059563 Index entries for sequences related to Beatty sequences %H A059563 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A059563 Beatty complement is A059564. %K A059563 nonn,easy %O A059563 1,1 %A A059563 Mitch Harris (maharri@cs.uiuc.edu), Jan 22, 2001 %I A022844 %S A022844 3,6,9,12,15,18,21,25,28,31,34,37,40,43,47,50,53,56,59,62,65,69,72, %T A022844 75,78,81,84,87,91,94,97,100,103,106,109,113,116,119,122,125,128, %U A022844 131,135,138,141,144,147,150,153,157,160,163,166,169,172,175,179 %N A022844 Beatty sequence for Pi. %H A022844 Index entries for sequences related to Beatty sequences %H A022844 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A022844 Cf. A054386. %K A022844 nonn %O A022844 0,1 %A A022844 Clark Kimberling (ck6@cedar.evansville.edu) %I A028251 %S A028251 3,6,9,12,15,18,22,25,28,31,35,38 %N A028251 Sequence arising in multiprocessor page migration. %D A028251 J Westbrook, Randomized algorithms for multiprocessor page migration, in "On-Line Algorithms", DIMACS Series in Discrete Math., Vol 7, 1992, pp. 135-149. %K A028251 nonn %O A028251 1,1 %A A028251 njas %I A039004 %S A039004 0,3,6,9,12,15,18,24,27,30,33,36,39,45,48,51,54,57,60,63,66,72,75,78, %T A039004 90,96,99,102,105,108,111,114,120,123,126,129,132,135,141,144,147,150, %U A039004 153,156,159,165,177,180,183,189,192,195,198,201,204,207,210,216,219 %N A039004 Representation in base 4 has same number of 1's and 2's. %K A039004 nonn,base,easy %O A039004 0,2 %A A039004 Olivier Gerard (ogerard@ext.jussieu.fr) %I A060293 %S A060293 0,1,3,6,9,12,15,19,22,26,30,34,38,42,46,50,55,59,63,68,72,77,82,86,91, %T A060293 96,101,106,110,115,120,125,130,135,141,146,151,156,161,166,172,177, %U A060293 182,188,193,198,204,209,215,220,225,231,236,242,248,253,259,264,270 %N A060293 Expected coupon collection numbers rounded up; i.e. if aiming to collect a set of n coupons, the expected number of random coupons required to receive the full set. %F A060293 a(n) =ceiling[n*sum_{1->n}(1/k)] =ceiling[n*A001008(n)/A002805(n)] =A052488(n)+1 for n>2 %e A060293 a(2)=3 since the prob of getting both coupons after two is 1/2, after 3 is 1/4, after 4 is 1/8, etc. and 2/2+3/2^2+4/2^3+.... =3. %K A060293 easy,nonn %O A060293 0,3 %A A060293 Henry Bottomley (se16@btinternet.com), Mar 24 2001 %I A061796 %S A061796 1,3,6,9,12,17,18,23,27,30,30,39,40,45,45,51,51,60,60,66,69,72,72,81, %T A061796 81,81,86,92,94,103,103,112,112,114,114,131,133,133,133,141,141,151, %U A061796 153,155,157,157,157,175,178,185,185,193,193,202,202,202,205,205,205 %N A061796 Number of distinct sums sigma(i) + sigma(j) for 1<=i<=j<=n, sigma(k) = A000203(k). %e A061796 If the {s+t} sums are generated by adding 2 terms of an S set consisting of n different entries, then at least 1, at most n(n+1)/2=A000217(n) distinct values can be obtained. The set of first n sigma-values gives results falling between these two extremes. E.g. n=10, A000203:{1,3,4,7,6,12,8,15,13,18...}. The 55 possible sigma(i)+sigma(j) additions give 30 different results:{2,4,5,6,...,33,36}. Therefore a(10)=30. %t A061796 f[x_] :=DivisorSigma[1,x] t0=Table[Length[Union[Flatten[Table[f[u]+f[w],{w,1,m},{u,1,m}]]]],{m,1,75}] %Y A061796 A000217, A000203. %K A061796 nonn %O A061796 1,2 %A A061796 Labos E. (labos@ana1.sote.hu), Jun 22 2001 %I A049707 %S A049707 1,1,3,6,9,12,17,22,27,33,39,48,56,64,71,82,92,105,116,129,139,153,167, %T A049707 183,197,213,227,247,262,285,303,325,342,362,379,403,426,451,471,497, %U A049707 520,551,576 %N A049707 a(n)=T(n,n+3), array T as in A049704. %K A049707 nonn %O A049707 0,3 %A A049707 Clark Kimberling, ck6@cedar.evansville.edu %I A052287 %S A052287 3,6,9,12,18,24,27,30,36,45,48,54,60,63,72,81,84,90,96,108,120 %N A052287 Start with 3; the general rule is "if x is present then so is xy for every y<=x". %F A052287 x is an element if and only if x=3*p1*p2*...*pk with primes 2<=p1<=p2<=...<=pk and 3*p1*p2*...*pi >= p(i+1) for all i < k. %e A052287 63 is an element because 63=3*3*7 and 3<=3 and 7<=3*3. %Y A052287 If instead we start with 2, we obtain the "Nullwertzahlen sequence" A047836. %K A052287 easy,nice,nonn %O A052287 0,1 %A A052287 Giuseppe Melfi (Giuseppe.Melfi@ima.unil.ch), Feb 08 2000 %I A063996 %S A063996 3,6,9,12,18,24,27,36,48,54,72,81,96,108,144,162,192,210,216,243,288, %T A063996 324,384,420,432,486,576,630,648,729,768,840,864,972,1050,1152,1260, %U A063996 1296,1458,1470,1536,1680,1728,1890,1944,2100,2187,2304,2520,2592,2916 %N A063996 n such that ud(n) = sopf(n)-1, where ud(n)=A034444(n) and sopf(n)=A008472(n). %o A063996 (PARI.2.0.17) sopf(n,s,fac,i)=fac=factor(n);for(i=1,matsize(fac)[1], s=s+fac[i,1]);return(s); ud(n) = 2^omega(n); j=[];for(n=1,7000,if(ud(n)==sopf(n)-1, j=concat(j,n)));j %Y A063996 Cf. A034444, A008472. %K A063996 easy,nonn %O A063996 1,1 %A A063996 Jason Earls (jcearls@kskc.net), Sep 06 2001 %I A065119 %S A065119 3,6,9,12,18,24,27,36,48,54,72,81,96,108,144,162,192,216,243,288,324, %T A065119 384,432,486,576,648,729,768,864,972,1152,1296,1458,1536,1728,1944, %U A065119 2187,2304,2592,2916 %N A065119 n-th cyclotomic polynomial is a trinomial. %e A065119 x^54-1 = (x^36+x^27-x^9-1) * (x^18-x^9+1) [second factor is cyclotomic and trinomial, so 54 is in the sequence] %p A065119 with(numtheory): a:=[]; for m from 1 to 3000 do if nops([coeffs(cyclotomic(m,x))])=3 then a:=[op(a),m] fi od; print(a); %Y A065119 Differs at the 18th term from A063996. %K A065119 nonn %O A065119 0,1 %A A065119 Len Smiley (smiley@math.uaa.alaska.edu), Nov 12 2001 %I A065811 %S A065811 0,3,6,9,12,45,48,63,66,78,111,123,126,138,150,231,243,261,309,324,348, %T A065811 351,363,402,414,417,441,456,504,570,582,660,684,765,768,849,873,951, %U A065811 963,1056,1125,1431,1518,1551,1638,1944,2013,2049,2139,2433,2457,2511 %N A065811 A065810[n]-1. %K A065811 nonn %O A065811 1,2 %A A065811 Antti.Karttunen@iki.fi Nov 22 2001 %I A061514 %S A061514 0,3,6,9,12,45,78,1011,4344,7677,1091010,43124343,76457676,10978109109, %T A061514 4312101143124312,7645434476457645,1097876771097810978, %U A061514 4312101110910104312101143121011,76454344431243437645434476454344 %N A061514 a(0) = 0; a(n) is obtained by incrementing each digit of a(n-1) by 3. %C A061514 In A061511-A061522, A061746-A061750 when the incremented digit exceeds 9 it is written as a 2-digit string. So 9+1 becomes the 2-digit string 10, etc. %K A061514 base,nonn,dumb %O A061514 0,2 %A A061514 Amarnath Murthy (amarnath_murthy@yahoo.com), May 08 2001 %E A061514 More terms from Larry Reeves (larryr@acm.org), May 11 2001 %I A022853 %S A022853 3,6,9,13,16,19,22,25,28,31,35,38,41,44,47,50,53,57,60,63,66,69,72, %T A022853 75,79,82,85,88,91,94,97,101,104,107,110,113,116,119,123,126,129, %U A022853 132,135,138,141,145,148,151,154,157,160,163,167,170,173,176,179 %N A022853 a(n) = integer nearest n*Pi. %K A022853 nonn %O A022853 0,1 %A A022853 Clark Kimberling (ck6@cedar.evansville.edu) %I A059540 %S A059540 3,6,9,13,16,19,22,26,29,32,35,39,42,45,48,52,55,58,61,65,68,71,75,78, %T A059540 81,84,88,91,94,97,101,104,107,110,114,117,120,123,127,130,133,136, %U A059540 140,143,146,150,153,156,159,163,166,169,172,176,179,182,185,189,192 %N A059540 Beatty sequence for 3^(1/3)/(3^(1/3)-1). %D A059540 Fraenkel, Aviezri S.; Levitt, Jonathan; Shimshoni, Michael; Characterization of the set of values f(n)=[n alpha], n=1,2,... Discrete Math.2 (1972),no.4,335-345. %H A059540 Index entries for sequences related to Beatty sequences %H A059540 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A059540 Beatty complement is A059539. %K A059540 nonn,easy %O A059540 1,1 %A A059540 Mitch Harris (maharri@cs.uiuc.edu), Jan 22, 2001 %I A059550 %S A059550 3,6,9,13,16,19,23,26,29,33,36,39,42,46,49,52,56,59,62,66,69,72,75,79, %T A059550 82,85,89,92,95,99,102,105,108,112,115,118,122,125,128,132,135,138, %U A059550 142,145,148,151,155,158,161,165,168,171,175,178,181,184,188,191,194 %N A059550 Beatty sequence for 1+ln(10). %D A059550 Fraenkel, Aviezri S.; Levitt, Jonathan; Shimshoni, Michael; Characterization of the set of values f(n)=[n alpha], n=1,2,... Discrete Math.2 (1972),no.4,335-345. %H A059550 Index entries for sequences related to Beatty sequences %H A059550 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A059550 Beatty complement is A059549. %K A059550 nonn,easy %O A059550 1,1 %A A059550 Mitch Harris (maharri@cs.uiuc.edu), Jan 22, 2001 %I A060605 %S A060605 1,3,6,9,13,16,20,24,28,32,37,41,46,50,55,60,66,70,75,80,85,90,96,101, %T A060605 107,112,117,122,128,133,139,145,151,157,163,168,174,179,185,191,198, %U A060605 203,209,215,221,227,234,240,246,252,259,265,272,277,284,290,296,302 %N A060605 The n-th term is the sum of lengths of iteration sequences of Euler totient function from 1 to n. %D A060605 P. Erdos, A. Granville, C. Pomerance and C. Spiro, On the normal behavior of the iterates of some arithmetic functions, in Analytic Number Theory, pp. 165-204. Birkhauser, Basel, 1990. %D A060605 H. Shapiro, An arithmetic function arising from Phi-function. American Math.Monthly 50:18-30. %F A060605 a(n)=Apply[Plus,{A049108(j),j=1..n}] %e A060605 Iteration sequences of Phi applied to 1, 2, 3, 4, 5, 6 give lengths 1, 2, 3, 3, 4, 3 with partial sums as follows:1, 3, 5, 9, 13, 16 resulting in 1st...6th terms here. %Y A060605 A049108, A003434. %K A060605 nonn %O A060605 0,2 %A A060605 Labos E. (labos@ana1.sote.hu), Apr 13 2001 %I A006590 M2522 %S A006590 1,3,6,9,13,16,21,24,29,33,38,41,48,51,56,61,67,70,77,80,87,92,97,100, %T A006590 109,113,118,123,130,133,142,145,152,157,162,167,177,180,185,190,199,202,211,214 %N A006590 Sum_{k=1..n} ceiling(n/k). %D A006590 M. Le Brun, personal communication. %K A006590 nonn,nice,easy %O A006590 2,2 %A A006590 njas %I A061781 %S A061781 1,3,6,9,13,17,21,25,29,33,39,44,50,54,59,63,67,75,80,86,91,95,101,107, %T A061781 114,120,126,131,136,140,148,154,160,168,174,180,187,192,199,205,211, %U A061781 219,224,231,237,242,249,255,264,270,278,283,289,296,302,306,310,319 %N A061781 Number of distinct sums p(i) + p(j) for 1<=i<=j<=n, p(k) = k-th prime. %F A061781 f[x_] :=Prime[x] Table[Length[Union[Flatten[Table[f[u]+f[w],{w,1,m},{u,1,m}]]]],{m,1,75}] %e A061781 If {p+q} sums are produced by adding 2 terms of an S set consisting of n different entries, then at least 1, at most n(n+1)/2=A000217(n) distinct values can be obtained. The set of first n primes gives results falling between these two extremes. E.g. S={2,3,5,7,11,13} provides 17 different sums of two, not necessarily different primes: {4,5,6,7,8,9,10,12,13,14,15,16,18,20,22,24,26}. Four sums arise more then once:10=3+7=5+5,14=3+11=7+7, 16=3+13=5+11,18=5+13=7+11. Thus a(6)=(6*7/2)-4=17. %Y A061781 Cf. A000217, A061784. %K A061781 nonn %O A061781 1,2 %A A061781 Labos E. (labos@ana1.sote.hu), Jun 22 2001 %I A002815 M2523 N0996 %S A002815 0,1,3,6,9,13,17,22,27,32,37,43,49,56,63,70,77,85,93,102,111,120,129,139, %T A002815 149,159,169,179,189,200,211,223,235,247,259,271,283,296,309,322,335, %U A002815 349,363,378,393,408,423,439,455,471 %N A002815 n + Sum_{k=1..n} pi(k), pi() = A000720. %D A002815 H. Brocard, Reply to Query 1421, Nombres premiers dans une suite de differences, L'Interm\'{e}diaire des Math\'{e}maticiens, 7 (1900), 135-137. %t A002815 Table[n + Sum[PrimePi[k], {k, 1, n}], {n, 0, 50}] %Y A002815 Cf. A000720. %K A002815 nonn,nice,easy %O A002815 0,3 %A A002815 njas, Robert G. Wilson v (rgwv@kspaint.com), mb %I A025205 %S A025205 0,1,3,6,9,13,17,22,27,33,39,46,53,60,68,76,84,93,102,111,120,130,140,151, %T A025205 161,172,183,195,206,218,230,242,255,267,280,293,306,320,334,347,361,375, %U A025205 390,404,419,434,449,464,479,495,510,526,542,558,574,591,607,624,640,657 %N A025205 a(n) = [ Sum of squares of log(k) ], k = 2,3,...,n. %K A025205 nonn %O A025205 2,3 %A A025205 Clark Kimberling (ck6@cedar.evansville.edu) %I A024190 %S A024190 1,3,6,9,13,17,22,27,33,39,46,53,60,69,77,86,96,106,117,128,140,152,165, %T A024190 178,191,206,220,235,251,267,284,301,319,337,356,375,394,415,435,456,478, %U A024190 500,523,546 %N A024190 [ (2nd elementary symmetric function of 3,4,...,n+3)/(3+4+...+n+3) ]. %K A024190 nonn %O A024190 1,2 %A A024190 Clark Kimberling (ck6@cedar.evansville.edu) %I A004116 M2524 %S A004116 1,3,6,9,13,17,22,27,33,39,46,53,61,69,78,87,97,107,118,129,141,153, %T A004116 166,179,193,207,222,237,253,269,286,303,321,339,358,377,397,417,438,459 %N A004116 [ (n^2 + 6n - 3)/4 ]. %D A004116 R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210. %H A004116 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 420 %K A004116 nonn %O A004116 1,2 %A A004116 njas %I A004129 M2525 %S A004129 1,3,6,9,13,17,22,27,33,40,47,56,65 %N A004129 Postage stamp problem. %D A004129 R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404. %K A004129 nonn %O A004129 2,2 %A A004129 njas %I A004137 M2526 %S A004137 3,6,9,13,17,23,29,36,43,50,59,60,79,90,101,112,123,138 %N A004137 Minimal nodes in graceful graph with n edges. %D A004137 Leech, John; On the representation of $1,2,\cdots,n$ by differences. J. London Math. Soc. 31 (1956), 160-169. %D A004137 J. C. P. Miller, Difference bases: Three problems in additive number theory, pp. 299-322 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971. %D A004137 J.-C. Bermond, Graceful graphs, radio antennae and French windmills, pp. 18-37 of R. J. Wilson, editor, Graph Theory and Combinatorics. Pitman, London, 1978. %Y A004137 Cf. A034470. %K A004137 nonn,nice,hard %O A004137 3,1 %A A004137 njas,sp %I A004131 M2527 %S A004131 1,3,6,9,13,17,24,30,36 %N A004131 Modular postage stamp problem. %D A004131 R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404. %D A004131 P Frankl et al., Projecting a finite point-set..., Ryukyu Math. J. 8 (1995), 27-35. %K A004131 nonn %O A004131 2,2 %A A004131 njas %I A032782 %S A032782 0,3,6,9,13,18,20,27,36,48,63,69,90,117,118,153,198,216,279,360,363, %T A032782 468,603,657,846,1098,1413,1818,1980,2547,3303,4248,5949,9918,12753, %U A032782 17856,29763,89298 %N A032782 n(n+1)(n+2)...(n+9) / n+(n+1)+(n+2)+...+(n+9) is an integer. %Y A032782 Cf. A032781, A032783. %K A032782 nonn %O A032782 0,2 %A A032782 Patrick De Geest (pdg@worldofnumbers.com), may 1998. %I A048202 %S A048202 3,6,9,13,18,24,29,31,37,43,47,51,54,61,67,69,71,74,78,82,88,94,97,99, %T A048202 103,110,117,122,126,129,135,147,157,161,164,168,171,176,181,183,192, %U A048202 204,211,215,224,235,243,251,259,265,270,278 %N A048202 a(n)=T(n,2), array T given by A048201. %K A048202 nonn %O A048202 2,1 %A A048202 Clark Kimberling, ck6@cedar.evansville.edu %I A034470 %S A034470 1,3,6,9,13,18,24,29,37,45 %N A034470 Leech's path-labeling problem for n nodes. %D A034470 Leech, John; On the representation of $1,2,\cdots,n$ by differences. J. London Math. Soc. 31 (1956), 160-169. %D A034470 R. K. Guy, Unsolved Problems in Number Theory, Sect. C10. %H A034470 Index entries for sequences related to trees %Y A034470 Cf. A007187. %K A034470 nonn,hard,nice %O A034470 2,2 %A A034470 njas %E A034470 a(12)>=51. %I A005488 M2528 %S A005488 3,6,9,13,18,24,29,37,45,51,61,70,79,93,101,113,127 %N A005488 Maximal edges in b^{hat} graceful graph with n nodes. %D A005488 J. C. P. Miller, Difference bases: Three problems in additive number theory, pp. 299-322 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971. %D A005488 J.-C. Bermond, Graceful graphs, radio antennae and French windmills, pp. 18-37 of R. J. Wilson, editor, Graph Theory and Combinatorics. Pitman, London, 1978. %K A005488 nonn %O A005488 3,1 %A A005488 njas,sp %I A014785 %S A014785 1,3,6,9,13,18,24,30,35,43,52,61,69,80,92,102,113,125,140, %T A014785 155,169,184,202,220,231,251,270,291,309,332,354,376,397, %U A014785 419,446,469,493,520,550,578,601,631,660,693,721,754,788 %N A014785 Sum [ k^2/n ], k=0..n. %D A014785 M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985, p. 103. %p A014785 f:=n->sum( ceil(k^2/n), k=0..n); %K A014785 nonn %O A014785 1,2 %A A014785 njas %I A033436 %S A033436 0,0,1,3,6,9,13,18,24,30,37,45,54,63,73,84,96,108,121,135, %T A033436 150,165,181,198,216,234,253,273,294,315,337,360,384,408, %U A033436 433,459,486,513,541,570,600,630,661,693,726,759,793,828 %N A033436 Number of edges in 4-partite Turan graph of order n. %D A033436 R. L. Graham et al., Handbook of Combinatorics, Vol 2, p. 1234. %F A033436 The second differences of the listed terms are periodic with period (1,1,1,0) of length 4, showing that the terms satisfy the recurrence a(n)=2a(n-1)-a(n-2)+a(n-4)-2a(n-5)+a(n-6) - John W. Layman (layman@math.vt.edu), Jan 23 2001 %K A033436 nonn,easy %O A033436 0,4 %A A033436 njas %I A002578 M2529 N0997 %S A002578 0,0,1,3,6,9,13,18,24,31 %N A002578 Integral points in a quadrilateral. %D A002578 Ehrhart, Eugene; Deux corollaires de la loi de reciprocite du polyedre rationnel. C. R. Acad. Sci. Paris Ser. A-B 265 1967 A160-A162. %Y A002578 Cf. A002579. %K A002578 nonn %O A002578 1,4 %A A002578 njas %I A059293 %S A059293 1,0,0,1,3,6,9,13,18,24,31,39,47,56,66,77,89,102,115,129,144,160, %T A059293 177,195,213,232,252,273,295,318,341,365,390,416,443,471,499,528, %U A059293 558,589,621,654,687,721,756,792,829,867,905,944,984,1025,1067 %N A059293 Round(n*(5*n-14)/12)+1. %D A059293 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 122, see #19 (2), I(n). %K A059293 nonn %O A059293 0,5 %A A059293 njas, Jan 25 2001 %I A000791 M2530 N0998 %S A000791 3,6,9,14,18,23,28,36 %N A000791 Ramsey numbers R(3,n). %D A000791 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 288. %D A000791 J. G. Kalbfleisch, Construction of special edge-chromatic graphs, Canad. Math. Bull., 8 (1965), 575-584. %D A000791 B. D. McKay, personal communication. %D A000791 H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 42. %D A000791 Jin Xu and C. K. Wong, Self-complementary graphs and Ramsey numbers I, Discrete Math., 223 (2000), 309-326. %H A000791 Stanislaw Radziszowski, Small Ramsey Numbers. %Y A000791 A row of table in A059442. %K A000791 nonn,hard,nice %O A000791 2,1 %A A000791 njas %E A000791 Next entry is in range 40-43. %I A027424 %S A027424 1,3,6,9,14,18,25,30,36,42,53,59,72,80,89,97,114,123,142,152,164, %T A027424 176,199,209,225,239,254,267,296,308,339,354,372,390,410,423,460, %U A027424 480,501,517,558,575,618,638,659,683,730,747,778,800,827,850,903 %N A027424 Number of distinct products ij with 1 <= i, j <= n (number of distinct terms in n X n multiplication table). %D A027424 C. Pomerance, "Paul Erdos,...", Notices Amer. Math. Soc., Vol. 45, Jan 1998, pp. 19-23. %Y A027424 Cf. A027384. %K A027424 nonn %O A027424 1,2 %A A027424 njas %I A058597 %S A058597 1,0,3,6,9,14,22,32,46,66,93,128,176,236,315,420,550,718,932,1198, %T A058597 1534,1956,2476,3120,3919 %N A058597 McKay-Thompson series of class 26B for Monster. %D A058597 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994). %Y A058597 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc. %K A058597 nonn %O A058597 -1,3 %A A058597 njas, Nov 27, 2000 %I A000741 M2531 N0999 %S A000741 1,3,6,9,15,18,27,30,45,42,66,63,84,84,120,99,153,132,174,165,231,180, %T A000741 270,234,297,270,378,276,435,360,450,408,540,414,630,513,636,552,780,558, %U A000741 861,690,828,759,1035,744,1113,870,1104,972,1326,945,1380,1116,1386,1218 %N A000741 Partitions of n into 3 unordered relatively prime parts. %D A000741 H. W. Gould, Binomial coefficients, the bracket function, and compositions with relatively prime summands, Fib. Quart. 2 (1964), 241-260. %H A000741 N. J. A. Sloane, Transforms %F A000741 Moebius transform of triangular numbers. %K A000741 nonn,easy %O A000741 3,2 %A A000741 njas %I A049991 %S A049991 0,1,3,6,9,15,19,25,33,41,47,60,67,77,92,104,113,132,142,158,178,193, %T A049991 205,231,247,264,289,310,325,359,375,397,427,449,473,513,532,556,591, %U A049991 623,644,689,711,741,788,817,841,892,920,957 %N A049991 a(n)=number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum <=n. %K A049991 nonn %O A049991 1,3 %A A049991 Clark Kimberling, ck6@cedar.evansville.edu %I A031940 %S A031940 1,3,6,9,15,19,28,33,45,51,66,73,91,99,120,129,153,163,190,201,231, %T A031940 243,276,289,325,339,378,393,435,451,496,513,561,579,630,649,703, %U A031940 780,801 %N A031940 Length of longest legal domino snake using full set of dominoes up to [ n:n ]. %H A031940 Index entries for sequences related to dominoes %F A031940 C(n,2)+1 if n odd, C(n,2)+n/2+1 if n even. %e A031940 E.g. for n=4 [ 1:1 ][ 1:2 ][ 2:2 ][ 2:3 ][ 3:3 ][ 3:1 ][ 1:4 ][ 4:4 ][ 4:2 ]. %Y A031940 Cf. A031878. %K A031940 nonn %O A031940 1,2 %A A031940 Colin L. Mallows (colinm@research.avayalabs.com) %I A007187 M2532 %S A007187 1,3,6,9,15,20,26,34,41 %N A007187 Leech's tree-labeling problem for n nodes. %D A007187 Leech, John; On the representation of $1,2,\cdots,n$ by differences. J. London Math. Soc. 31 (1956), 160-169. %D A007187 R. K. Guy, A quarter century of "Monthly" unsolved problems, 1969-1993, Amer Math. Monthly, 100 (1993), 945-949. %D A007187 R. K. Guy, Unsolved Problems in Number Theory, Sect. C10. %H A007187 Index entries for sequences related to trees %Y A007187 Cf. A034470. %K A007187 nonn,hard,nice %O A007187 2,2 %A A007187 njas %E A007187 a(11)>=48, a(12)>=55. %I A002597 M2533 N1000 %S A002597 1,1,3,6,9,15,25,34,51,73,97,132,178,226,294,376,466,582,722,872,1062, %T A002597 1282,1522,1812,2147,2507,2937,3422,3947,4557,5243,5978,6825,7763,8771, %U A002597 9912,11172,12516,14028,15680,17444,19404,21540,23808,26316,29028 %N A002597 A generalized partition function. %D A002597 Gupta, Hansraj; A generalization of the partition function. Proc. Nat. Inst. Sci. India 17, (1951). 231-238. %F A002597 Generating function: 1/((1-x)*(1-x^2)^2*(1-x^3)^3) %K A002597 nonn %O A002597 0,3 %A A002597 njas %E A002597 More terms and formula from Henry Bottomley (se16@btinternet.com), Sep 17 2001 %I A057855 %S A057855 1,3,6,9,16,21,30,36,46,61,68,86,99,110,126,146,168,184,205,223,242, %T A057855 270,292,321,360,381,404,429,446,477,546,574,614,637,693,717,762,804, %U A057855 842,890,935,965,1029,1059,1105,1134,1222,1304,1348,1381,1423,1483 %N A057855 Greatest k such that (kth prime) <= (n times n-th prime). %C A057855 Might be roughly n^2/2 (seems to be marginally more at least for small n) %e A057855 a(4)=9 since 4th prime is 7, 4*7=28, greatest prime less than or equal to 28 is 23 which is the 9th prime. %t A057855 a(n) = Pi[n.p(n)] Table[PrimePi[w*Prime[w]],{w,1,100}] %Y A057855 Cf. A020900, A020901, A020934-A020940. %K A057855 nonn %O A057855 1,2 %A A057855 Henry Bottomley (se16@btinternet.com), Nov 13 2000 %I A047847 %S A047847 1,3,6,9,18,21,33,39,48,51,54,63,81,96,111,114,138,153,156,174,189,198, %T A047847 219,228,231,243,249,306,321,336,369,378,384,411,426,429,438,441,453, %U A047847 468,483,504,543,546,606,639,648,651,711,714,723,741,744,774,783,789 %N A047847 n + (n+1) and (n+2) + (n+3) both prime. %F A047847 a(n) =(A046132(n)-5)/2 =(A023200(n)-1)/2 =3*A056956(n-1) %e A047847 If n = 6, then 6 + 7 = 13 and 8 + 9 = 17 are both prime. %Y A047847 Cf. A005097, A014545. %K A047847 easy,nonn %O A047847 1,2 %A A047847 Enoch Haga (EnochHaga@msn.com) %E A047847 Corrected by Henry Bottomley (se16@btinternet.com), Jul 18 2000 %I A007783 %S A007783 3,6,9,18,22,32,46,58,77,97,114,135,160,186,218 %N A007783 Van der Waerden numbers W(2;2,n-1). %D A007783 M. D. Beeler and P. E. O'Neil, Some new Van der Waerden numbers, Discrete Math., 28 (1979), 135-146. %D A007783 V. Chvatal, Some unknown Van der Waerden numbers, pp. 31-33 of R. K. Guy et al., editors, Combinatorial Structures and Their Applications (Proceedings Calgary Conference June 1969}), Gordon and Breach, NY, 1970. %Y A007783 Cf. A002886. %K A007783 nonn,hard %O A007783 0,1 %A A007783 Matthew Klimesh [ matthew@engin.umich.edu ] %I A050625 %S A050625 3,6,9,18,27,36,45,54,63,72,81,90,99,108,135,162,189,216,243,270,297, %T A050625 324,351,378,405,432,459,486,513,540,567,594,621,648,675,702,729,756, %U A050625 783,810,837,864,891,918,945,972,999,1053,1134,1215,1296,1377,1458 %N A050625 Divisible by 3^k (where k is digit length of a(n)). %Y A050625 Cf. A035014, A050622, A050623. %K A050625 nonn,base,easy %O A050625 1,1 %A A050625 Patrick De Geest (pdg@worldofnumbers.com), Jun 1999. %I A025614 %S A025614 1,3,6,9,18,27,36,54,81,108,162,216,243,324,486,648,729,972,1296,1458, %T A025614 1944,2187,2916,3888,4374,5832,6561,7776,8748,11664,13122,17496,19683, %U A025614 23328,26244,34992,39366,46656,52488,59049,69984,78732,104976,118098 %N A025614 Numbers of form 3^i*6^j, with i, j >= 0. %K A025614 easy,huge,nonn %O A025614 1,2 %A A025614 dww %I A057576 %S A057576 3,6,9,18,28,54,108 %N A057576 Maximal numbers of codewords in mixed code with n binary coordinates and 3 ternary coordinates with Hamming distance 3. %D A057576 P. R. J. Ostergard, Classification of binary/ternary one-error-correcting codes, Discrete Math., 223 (2000), 253-262. %Y A057576 Cf. A057574-A057584, A050142. %K A057576 nonn %O A057576 0,1 %A A057576 njas, Oct 04 2000 %I A059006 %S A059006 3,6,9,18,65,66,287,354 %N A059006 x^n + x^2 + 1 is irreducible over GF(7). %t A059006 Do[ If[ ToString[ Factor[ x^n + x^2 + 1, Modulus -> 7 ] ] == ToString[ x^n + x^2 + 1 ], Print[ n ] ], {n, 1, 1016} ] %K A059006 nonn %O A059006 1,1 %A A059006 Robert G. Wilson v (rgwv@kspaint.com), Jan 16 2001 %I A018186 %S A018186 1,3,6,9,19,30,63,99,208,327,687,1080,2269,3567,7494,11781, %T A018186 24751,38910,81747,128511,269992,424443,891723,1401840, %U A018186 2945161,4629963,9727206,15291729,32126779,50505150,106107543 %N A018186 a_{n+2}=3a_n-a_{n-2}, for n\ge2. %D A018186 J.L. Simons, {\it Conditional recurring sequences}, Doctor's Thesis, Delft University of Technology, Delft, 1976 ({\bf MR} 54 \#7361) ]. %F A018186 G.f.: (1+3x+3x^2)/(1-3x^2-x^4). %K A018186 nonn %O A018186 0,2 %A A018186 H. J. J. te Riele (Herman.te.Riele@cwi.nl) %I A015938 %S A015938 1,3,6,9,20,56,66,133,260,513,1030,2091,4128,8593,16394,33195,65584, %T A015938 131345,262176,524989,1048660,2097291,4195642,8388997,16777272, %U A015938 33554525,67109198,134217729,268435468,536875753,1073741910 %N A015938 First k such that k | 2^k - 2^n. %K A015938 nonn %O A015938 0,2 %A A015938 Robert G. Wilson v (rgwv@kspaint.com) %I A050889 %S A050889 1,3,6,9,21,34,43,49,114,115,178,322,411,438,454,589,598,754,801,1173, %T A050889 1546,1699,2155,2994,4038,5439,6442,10699,12274,13314,13911 %N A050889 261*2^n-1 is prime. %H A050889 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A050889 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A050889 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %K A050889 hard,nonn %O A050889 0,2 %A A050889 njas, Dec 29 1999 %I A026095 %S A026095 1,3,6,9,22,55,153,431,1235,3555,10278,29793,86562,251983,734776,2145825, %T A026095 6275148,18373299,53856156,158025189,464112300,1364247183,4013353935, %U A026095 11815188003,34807249137,102606325139,302646363728,893175905781 %N A026095 a(n) = SUM{T(k,k-1)}, k = 1,2,...,n, where T is the array defined in A026082. %K A026095 nonn %O A026095 1,2 %A A026095 Clark Kimberling (ck6@cedar.evansville.edu) %I A061929 %S A061929 1,3,6,9,24,28,27,90,132,120,81,324,576,624,496,243,1134,2376,3024, %T A061929 2736,2016,729,3888,9396,13824,14256,11520,8128,2187,13122,35964,60264, %U A061929 70416,63072,47424,32640,6561,43740,134136,252720,331776,330048,268416 %N A061929 Triangle with n >= k >= 0 where a(n,k) = 2^k*3^(n-k)*(C(n+1,0)+C(n+1,1)+...C(n+1,k)). %F A061929 a(n,k) =A054143(n,k)*A036561(n,n-k) %e A061929 Rows start (1), (3,6), (9,24,68), (27,90,132,120) etc. %Y A061929 Row sums are 5^(n+1)-4^(n+1), i.e. A005060. Cf. A061930. %K A061929 nonn,tabl %O A061929 0,2 %A A061929 Henry Bottomley (se16@btinternet.com), May 22 2001 %I A019461 %S A019461 0,1,1,3,6,9,27,31,124,129,645,651,3906,3913,27391,27399, %T A019461 219192,219201,1972809,1972819,19728190,19728201,217010211, %U A019461 217010223,2604122676,2604122689,33853594957,33853594971 %N A019461 Add 1, multiply by 1, add 2, multiply by 2, etc. %p A019461 A019461:=proc(n) option remember; if n = 0 then 1 elif n mod 2 = 1 then floor( (n+1)/2 )+A019461(n-1) else floor( (n+1)/2 )*A019461(n-1); fi; end; %K A019461 nonn %O A019461 0,4 %A A019461 njas %I A045638 %S A045638 3,6,9,33,66,99,111,141,171,222,252,282,303,333,363,393,414,444,474, %T A045638 525,555,585,606,636,666,696,717,747,777,828,858,888,909,939,969,999, %U A045638 1221,1551,1881,2112,2442,2772,3003,3333,3663,3993,4224,4554,4884,5115 %N A045638 Palindromic and divisible by 3. %K A045638 nonn %O A045638 0,1 %A A045638 Jeff Burch (gburch@erols.com) %I A038224 %S A038224 1,3,6,9,36,36,27,162,324,216,81,648,1944,2592,1296,243,2430,9720, %T A038224 19440,19440,7776,729,8748,43740,116640,174960,139968,46656,2187, %U A038224 30618,183708,612360,1224720,1469664,979776,279936,6561,104976 %N A038224 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*6^j. %D A038224 B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct. 1996), pp. 109-121. %K A038224 nonn,tabl,easy %O A038224 0,2 %A A038224 njas %I A057241 %S A057241 0,0,3,6,10 %N A057241 Circuit cost of hardest Boolean functions of n variables; metric: 2-input AND-gates cost 1, NOT is free, fanout is free, inputs are free, no feedback allowed. %H A057241 Comments and examples %Y A057241 Cf. A056287, A058759. %K A057241 nonn,nice,bref,hard %O A057241 0,3 %A A057241 Richard Schroeppel (rcs@CS.Arizona.EDU), Jan 10 2001 %E A057241 a(5) <= 23. %I A065234 %S A065234 1,1,3,6,10,5,11,18,26,8,18,29,41,2,16,31,47,64,82,16,36,57,79,102,126, %T A065234 25,51,78,106,135,165,21,53,86,120,155,191,228,34,73,113,154,196,239, %U A065234 283,31,77,124,172,221,271,322,4,57,111,166,222,279,337,396,5,66,128 %N A065234 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the decagonal numbers. The first elements of the rows form a(n). %Y A065234 Cf. A064766, A064865, A065221-A065233. %K A065234 easy,nonn %O A065234 0,3 %A A065234 Floor van Lamoen (f.v.lamoen@wxs.nl), Oct 22 2001 %I A055262 %S A055262 0,1,3,6,10,6,12,10,9,18,19,21,15,19,24,21,19,27,27,28,30,24,28,33,30, %T A055262 28,36,36,37,39,42,37,42,39,46,45,45,46,48,51,46,51,48,55,54,54,55,57, %U A055262 60,55,60,57,64,63,63,64,66,69,73,69,75,73,72,72,73,75,78,82,78,84,82 %N A055262 n + sum of digits of a(n-1). %F A055262 a(n) = n+A055263(n-1) =n+A007953(a(n-1)) %e A055262 a(10)=19 because a(9)=18, 1+8=9 and 10+9=19 %Y A055262 Cf. A055263. %K A055262 base,easy,nonn %O A055262 0,3 %A A055262 Henry Bottomley (se16@btinternet.com), May 08 2000 %I A009019 %S A009019 1,0,1,3,6,10,6,77,724,5220,36364,259457,1933704,15161250,125278984, %T A009019 1090268137,9973751184,95650827080,958577301104,10001308630117, %U A009019 108164953941664,1206196891901710,13774788612312224 %V A009019 1,0,-1,3,-6,10,-6,-77,724,-5220,36364,-259457,1933704,-15161250,125278984, %W A009019 -1090268137,9973751184,-95650827080,958577301104,-10001308630117, %X A009019 108164953941664,-1206196891901710,13774788612312224 %N A009019 Expansion of cos(ln(1+sin(x))). %t A009019 Cos[ Log[ 1+Sin[ x ] ] ] %K A009019 sign,done,easy %O A009019 0,4 %A A009019 rhh@research.bell-labs.com %E A009019 Extended with signs 03/97 by Olivier Gerard. %I A032570 %S A032570 1,1,1,1,1,1,1,1,1,3,6,10,10,10,10,10,13,12,13,12,13,13,13,13,13,13,24, %T A032570 19,19,19,19,19,23,22,21,22,41,40,46,46,46,46,46,46,73,79,76,88,160, %U A032570 166,157,163,304,307,316,562,502,496,594,545,580,604,590,602,1108 %N A032570 Quotient of 'base 19' division described in A032569. %Y A032570 Cf. A032569. See also A032563 for explanation. %K A032570 nonn,hard %O A032570 0,10 %A A032570 Patrick De Geest (pdg@worldofnumbers.com), april 1998. %E A032570 More terms from Larry Reeves (larryr@acm.org), Oct 02 2000 %E A032570 The next term has an A032569 value > 2.4*10^11 %I A043321 %S A043321 3,6,10,11,12,15,19,20,21,24,31,32,34,35,37,38,39,42,46,47,48, %T A043321 51,58,59,61,62,64,65,66,69,73,74,75,78,94,95,97,98,103,104,106, %U A043321 107,112,113,115,116,118,119,120,123,127,128,129,132,139,140 %N A043321 Number of 0's in base 3 is 1. %K A043321 nonn,base %O A043321 1,1 %A A043321 Clark Kimberling, ck6@cedar.evansville.edu %I A050107 %S A050107 3,6,10,12,13,16,18,22,23,26,27,31,33,35,37,39,41,44,46,51,53,54,57,59, %T A050107 60,63,66,67,69,71,74,76,81,83,84,86,88,90,93,95,97,99,102,105,107,109, %U A050107 111,113,116,118,121,122,125,126,129,132 %N A050107 a(n) = n-th number k such that b(k)>b(k+1), where b=A050104. %K A050107 nonn %O A050107 1,1 %A A050107 Clark Kimberling, ck6@cedar.evansville.edu %I A015875 %S A015875 3,6,10,12,14,15,22,24,26,30,34,42,44,46,62,69,74,82,94,106,114,118, %T A015875 122,134,135,146,166,194,198,202,206,214,262,282,302,314,334,346, %U A015875 366,382,386,446,454,466,474,502,514,526,542,554,614,622,662,694 %N A015875 Phi(n + 12) | sigma(n) + 12. %K A015875 nonn %O A015875 0,1 %A A015875 Robert G. Wilson v (rgwv@kspaint.com) %I A007960 %S A007960 1,3,6,10,12,15,18,19,21,28,30,33,36,37,39,45,46,51,54,55,57,60,63, %T A007960 66,73,75,78,81,82,87,91,93,99,100,102,105,108,109,111,117,118,120 %N A007960 Some permutation of digits is a triangular number. %D A007960 F. Smarandache, "Only Problems, not Solutions!", Xiquan Publ., Phoenix-Chicago, 1993. %H A007960 M. L. Perez et al., eds., Smarandache Notions Journal %K A007960 nonn,base %O A007960 1,2 %A A007960 R. Muller %E A007960 Corrected a large number of errors 4/15/96. I'm not sure how rigorous this is - to prove that 17 (say) is missing, one would have to prove that there is no triangular number consisting of a 1, a 7 and any number of 0's. %I A032732 %S A032732 0,1,3,6,10,12,15,21,22,30,34,36,45,48,49,52,54,55,58,63,64,66,69,75, %T A032732 78,82,87,91,94,103,105,106,108,111,112,120,124,132,133,135,138,139, %U A032732 141,142,147,154,156,169,171,178,180,184,187,189,190,199,201,208,210 %N A032732 n prefixed by '7' and followed by '9' is a prime. %K A032732 nonn,base %O A032732 0,3 %A A032732 Patrick De Geest (pdg@worldofnumbers.com), may 1998. %I A028357 %S A028357 1,3,6,10,13,15,16,18,21,25,28,30,31,33,36,40,43,45,46, %T A028357 48,51,55,58,60,61,63,66,70,73,75,76,78,81,85,88,90,91, %U A028357 93,96,100,103,105,106,108,111,115,118,120,121,123,126 %N A028357 Partial sums of A028356. %K A028357 nonn %O A028357 0,2 %A A028357 njas %I A028433 %S A028433 3,6,10,13,16,19,22,25,87,96,105,114,123,132,141,150,159,168,177,186, %T A028433 195,204,213,222,231,240,739,766,793,820,847,874,901,928,955,982,1009, %U A028433 1036,1063,1090,1117,1144,1171,1198,1225,1252,1279,1306,1333,1360 %N A028433 Golc sequence in base 3. Left to right concatenation of n,int(log_3(n)),int(log_3(int(log_3(n)))),... in base 3. %K A028433 nonn,base,easy %O A028433 1,1 %A A028433 Olivier Gerard (ogerard@ext.jussieu.fr) %I A001952 M2534 N1001 %S A001952 3,6,10,13,17,20,23,27,30,34,37,40,44,47,51,54,58,61,64,68,71,75,78, %T A001952 81,85,88,92,95,99,102,105,109,112,116,119,122,126,129,133,136,139, %U A001952 143,146,150,153 %N A001952 A Beatty sequence: [ n * (2 + sqrt(2)) ]. %D A001952 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Pellian representatives, Fib. Quart., 10 (1972), 449-488. %D A001952 I. G. Connell, A generalization of Wythoff's game, Canad. Math. Bull., 2 (1959), 181-190. %D A001952 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 77. %H A001952 Index entries for sequences related to Beatty sequences %H A001952 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A001952 Cf. A001951. %K A001952 nonn,easy,nice %O A001952 1,1 %A A001952 njas %I A047280 %S A047280 3,6,10,13,17,20,24,27,31,34,38,41,45,48,52,55,59,62,66,69,73,76,80,83, %T A047280 87,90,94,97,101,104,108,111,115,118,122,125,129,132,136,139,143,146, %U A047280 150,153,157,160,164,167,171 %N A047280 Congruent to {3, 6} mod 7. %K A047280 nonn %O A047280 0,1 %A A047280 njas %I A049880 %S A049880 1,3,6,10,13,17,21,25,29,35,39,45,50,54,59,63,70,75,81,86,91,97,102, %T A049880 109,114,119,125,130,135,143,148,154,162,168,175,181,187,195,200,206, %U A049880 213,218,224,230,236,242,249,258,263 %N A049880 a(n)=number of distinct sums of 2 of the first n primes. %K A049880 nonn %O A049880 2,2 %A A049880 Clark Kimberling, ck6@cedar.evansville.edu %I A027428 %S A027428 0,1,3,6,10,13,19,24,31,36,46,51,63,70,78,87,103,111,129,139,150, %T A027428 161,183,192,210,223,239,252,280,291,321,337,354,371,390,403,439, %U A027428 458,478,493,533,549,591,611,631,654,700,717,752,774,800,823,875 %N A027428 Number of distinct products ij with 1 <= i < j <= n. %K A027428 nonn,easy,nice %O A027428 1,3 %A A027428 njas %I A022764 %S A022764 1,3,6,10,14,16,23,27,35,37,42,45,51,57,65,70,75,81,87,94,101,105, %T A022764 111,119,126,133,138,147,153,156,162,168,171,177,181,189,198,204, %U A022764 207,214,220,231,243,248,255,264,270,277,284,286,296,309,314,316 %N A022764 (n-th 8k+3 prime plus n-th 8k+5 prime)/8. %K A022764 nonn %O A022764 1,2 %A A022764 Clark Kimberling (ck6@cedar.evansville.edu) %I A036572 %S A036572 3,6,10,14,19,24,30,36,43,50 %N A036572 Number of tetrahedra in largest triangulation of polygonal prism with regular polygonal base. %H A036572 J.A. De Loera, Computing minimal and maximal triangulations of polytopes. %Y A036572 Cf. A036573. %K A036572 nonn %O A036572 3,1 %A A036572 Jesus De Loera (deloera@math.ucdavis.edu) %I A033437 %S A033437 0,0,1,3,6,10,14,19,25,32,40,48,57,67,78,90,102,115,129, %T A033437 144,160,176,193,211,230,250,270,291,313,336,360,384,409, %U A033437 435,462,490,518,547,577,608,640,672,705,739,774,810,846 %N A033437 Number of edges in 5-partite Turan graph of order n. %D A033437 Graham et al., Handbook of Combinatorics, Vol 2, p. 1234. %K A033437 nonn %O A033437 0,4 %A A033437 njas %I A024928 %S A024928 3,6,10,14,20,24,30,34,41,46,54,59,70,76,82,88,96,102,110,115,124,131,139, %T A024928 147,156,162,170,179,192,197,209,219,228,236,246,253,265,271,280,288,299, %U A024928 305,316,323,332,341,352,358,370,377,385,392,406,413,425,431,440,449,462 %N A024928 Sum of [ (n + p(k))/k ], k = 1,2,3,...,n. %K A024928 nonn %O A024928 1,1 %A A024928 Clark Kimberling (ck6@cedar.evansville.edu) %I A025206 %S A025206 0,0,1,3,6,10,14,20,27,35,45,55,67,81,96,112,129,148,169,191,214,239,266, %T A025206 294,324,355,388,422,458,496,535,576,618,662,708,756,805,855,908,962,1018, %U A025206 1076,1135,1196,1258,1323,1389,1457,1526,1598,1671,1746,1822,1900,1980,2062 %N A025206 [ Sum{(log(j)-log(i))^2} ], 2 <= i < j <= n. %K A025206 nonn %O A025206 3,4 %A A025206 Clark Kimberling (ck6@cedar.evansville.edu) %I A049989 %S A049989 1,3,6,10,14,21,26,33,42,51,58,72,80,91,107,120,130,150,161,178,199, %T A049989 215,228,255,272,290,316,338,354,389,406,429,460,483,508,549,569,594, %U A049989 630,663,685,731,754,785,833,863,888,940,969 %N A049989 a(n)=number of arithmetic progressions of positive integers, nondecreasing with sum <= n. %K A049989 nonn %O A049989 1,2 %A A049989 Clark Kimberling, ck6@cedar.evansville.edu %I A054731 %S A054731 0,3,6,10,15,18,21,28,30,36,45,55,60,63,66,78,84,90,91,105,108,120,126, %T A054731 135,136,150,153,165,168,171,190,198,210,216,231,234,253,270,273,276, %U A054731 280,300,315,325,330,351,360,378,396,406,408,420,435,450,459,465,468 %N A054731 Numbers of the form x*(x + 1)*y*(y + 1)/4 where x and y are distinct. %Y A054731 Cf. A053990, A054734. Contains all triangular numbers >1. %K A054731 nonn,easy %O A054731 1,2 %A A054731 stuart m. ellerstein (ellerstein@aol.com), Apr 22 2000 %E A054731 More terms from James A. Sellers (sellersj@math.psu.edu), Apr 22 2000 %I A029716 %S A029716 1,3,6,10,15,18,25,29,35,40,51,55,68,75,80,86,103,109,128, %T A029716 133,140,151,174,178,188,201,210,217,246,251,282,290,301, %U A029716 318,325,331,368,387,400,405,446,453,496,507,513,536,583 %N A029716 Partial sums of Smarandache's function A002034. %K A029716 nonn %O A029716 0,2 %A A029716 njas %I A027920 %S A027920 3,6,10,15,20,26,33,40,49,58,68,78,90,102,115,129,143,158,174, %T A027920 191,208,227,246,265,286,307,329,352,376,400,425,451,477,505, %U A027920 533,562,591 %N A027920 Least k such that 2nd elementary symmetric function of {1,2,...,k} >= 4th elementary symmetric function of {1,2,...,n}. %K A027920 nonn %O A027920 4,1 %A A027920 Clark Kimberling, ck6@cedar.evansville.edu %I A033438 %S A033438 0,0,1,3,6,10,15,20,26,33,41,50,60,70,81,93,106,120,135, %T A033438 150,166,183,201,220,240,260,281,303,326,350,375,400,426, %U A033438 453,481,510,540,570,601,633,666,700,735,770,806,843,881 %N A033438 Number of edges in 6-partite Turan graph of order n. %D A033438 Graham et al., Handbook of Combinatorics, Vol 2, p. 1234. %K A033438 nonn %O A033438 0,4 %A A033438 njas %I A047800 %S A047800 1,3,6,10,15,20,27,34,42,51,61,71,83,94,106,120,135,148,165,180,198, %T A047800 216,235,252,273,294,315,337,360,382,408,431,457,484,508,536,567, %U A047800 595,624,653,687,715,749,781,813,850,884,919,957,993 %N A047800 Number of different values of i^2+j^2 for i,j in [ 0,n ]. %t A047800 Table[ Length@Union[ Flatten[ Table[ i^2+j^2,{i,0,n},{j,0,n} ] ] ],{n,0,49} ] %Y A047800 Cf. A034966, A047801. %K A047800 nonn,easy,nice %O A047800 0,2 %A A047800 w.meeussen.vdmcc@vandemoortele.be %I A034175 %S A034175 0,1,3,6,10,15,21,4,5,11,14,2,7,9,16,20,29,35,46,18,31,33,48,52,12,13, %T A034175 23,26,38,43,57,24,25,39,42,22,27,37,44,56,8,17,19,30,34,47,53,28,36, %U A034175 45,55,66,78,91,105,64,80,41,40,60,61,83,86,58,63,81,88,108,117,79,65 %N A034175 a(n) is minimal such that a(n)+a(n-1) is a square and a(n) is not in {a(0), ..., a(n-1)}. %C A034175 Conjectured to be a permutation of the nonnegative integers. %H A034175 Index entries for sequences that are permutations of the natural numbers %t A034175 a[ 0 ]=b[ 0 ]=0; lst={0}; For[ n=1,n<250,n++,For[ s=Ceiling[ Sqrt[ a[ n-1 ] ] ],MemberQ[ lst,s^2-a[ n-1 ] ],s++,Null ];b[ a[ n ]=s^2-a[ n-1 ] ]=n;AppendTo[ lst,a[ n ] ] ]; Table[ a[ n ],{n,0,100} ] %Y A034175 Cf. A064928, A064929, A064930. %K A034175 nonn,easy,nice %O A034175 0,3 %A A034175 Dean Hickerson (dean@math.ucdavis.edu) %I A056150 %S A056150 1,3,6,10,15,21,25,27,25,21,15,10,6,3,1 %N A056150 Number of combinations for each possible sum when throwing 3 (normal) dice. %e A056150 Using three normal (six-sided) dice we can produce a sum of 3 in only one way: 1,1,1. We can produce a sum of 4 in three ways: 1,1,2; 1,2,1; 2,1,1. We can produce a sum of 5 in 6 ways, and so on. %K A056150 nonn,fini,full %O A056150 0,2 %A A056150 Joe Slater (joe@yoyo.cc.monash.edu.au), Aug 05 2000 %I A033439 %S A033439 0,0,1,3,6,10,15,21,27,34,42,51,61,72,84,96,109,123,138, %T A033439 154,171,189,207,226,246,267,289,312,336,360,385,411,438, %U A033439 466,495,525,555,586,618,651,685,720,756,792,829,867,906 %N A033439 Number of edges in 7-partite Turan graph of order n. %D A033439 Graham et al., Handbook of Combinatorics, Vol 2, p. 1234. %K A033439 nonn %O A033439 0,4 %A A033439 njas %I A061786 %S A061786 1,3,6,10,15,21,27,34,42,52,61,72,83,94,108,122,135,151,165,183,200, %T A061786 218,234,254,275,296,317,339,361,387,409,434,460,484,512,542,570,598, %U A061786 627,661,689,722,753,784,821,854,888,925,960,998,1036,1075,1109,1148 %N A061786 Number of distinct sums i^2 + j^2 for 1<=i<=j<=n. %e A061786 If the {s+t} sums are generated by addition 2 terms of an S set consisting of n different entries, then at least 1, at most n(n+1)/2=A000217(n) distinct values can be obtained. The set of first n squares gives results falling between these two extremes. E.g. S={1,4,9,16,25,36,49} provides 27 different sums of two, not necessarily different squares: {2,5,8,10,13,17,18,20,25,26,29,32,34,37,40,41,45,50,52,53,58,61,65,72,74,85,98}_ Only a single sum arises more then once: 50=1+49=25+25. Therefore a(7)=(7*8/2)-1=27. %t A061786 f[x_] :=x^2 Table[Length[Union[Flatten[Table[f[u]+f[w],{w,1,m},{u,1,m}]]]],{m,1,75}] %Y A061786 A000217. %K A061786 nonn %O A061786 1,2 %A A061786 Labos E. (labos@ana1.sote.hu), Jun 22 2001 %I A054636 %S A054636 0,1,3,6,10,15,21,28,29,29,30,31,32,34,35,38,39,43,44,49,50,56, %T A054636 57,64,66,66,68,69,71,73,75,78,80,84,86,91,93,99,101,108,111,111, %U A054636 114,115,118,120,123,126,129,133,136,141,144,150,153,160,164,164 %N A054636 Partial sums of A054634. %K A054636 nonn %O A054636 0,3 %A A054636 njas, Apr 16 2000 %I A033440 %S A033440 0,0,1,3,6,10,15,21,28,35,43,52,62,73,85,98,112,126,141, %T A033440 157,174,192,211,231,252,273,295,318,342,367,393,420,448, %U A033440 476,505,535,566,598,631,665,700,735,771,808,846,885,925 %N A033440 Number of edges in 8-partite Turan graph of order n. %D A033440 Graham et al., Handbook of Combinatorics, Vol 2, p. 1234. %K A033440 nonn %O A033440 0,4 %A A033440 njas %I A033441 %S A033441 0,0,1,3,6,10,15,21,28,36,44,53,63,74,86,99,113,128,144, %T A033441 160,177,195,214,234,255,277,300,324,348,373,399,426,454, %U A033441 483,513,544,576,608,641,675,710,746,783,821,860,900,940 %N A033441 Number of edges in 9-partite Turan graph of order n. %D A033441 Graham et al., Handbook of Combinatorics, Vol 2, p. 1234. %K A033441 nonn %O A033441 0,4 %A A033441 njas %I A061076 %S A061076 1,3,6,10,15,21,28,36,45,45,46,48,51,55,60,66,73,81,90,90,92,96,102, %T A061076 110,120,132,146,162,180,180,183,189,198,210,225,243,264,288,315,315, %U A061076 319,327,339,355,375,399,427,459,495,495,500,510,525,545,570,600,635 %N A061076 a(n) is the sum of the products of the digits of all the numbers from 1 to n. %D A061076 Amarnath Murthy, Smarandaceh friendly numbers and a few more sequences, Smarandache Notions Journal, Vol.12, No.1-2-3, Spring 2001. %H A061076 M. L. Perez et al., eds., Smarandache Notions Journal %F A061076 a(n) = Sum_{k = 1..n} product of the digits of k. %e A061076 a(9)=a(10)=1+2+3+4+5+6+7+8+9+1x0=1+2+3+4+5+6+7+8+9=45. %K A061076 nonn,base,easy %O A061076 0,2 %A A061076 Amarnath Murthy (amarnath_murthy@yahoo.com), Apr 14 2001 %E A061076 Corrected and extended by Matthew M. Conroy (doctormatt@earthlink.net), Apr 16 2001 %I A054632 %S A054632 0,1,3,6,10,15,21,28,36,45,46,46,47,48,49,51,52,55,56,60,61,66,67, %T A054632 73,74,81,82,90,91,100,102,102,104,105,107,109,111,114,116,120, %U A054632 122,127,129,135,137,144,146,154,156,165,168,168,171,172,175,177 %N A054632 Partial sums of A007376. %K A054632 nonn,base,easy %O A054632 0,3 %A A054632 njas, Apr 16 2000 %I A037123 %S A037123 1,3,6,10,15,21,28,36,45,46,48,51,55,60,66,73,81,90,100,102,105,109, %T A037123 114,120,127,135,144,154,165,168,172,177,183,190,198,207,217,228,240, %U A037123 244,249,255,262,270,279,289,300,312,325,330,336,343,351 %N A037123 a(n) = a(n-1) + Sum of digits of n. %Y A037123 Cf. A004207, A016052. %K A037123 nonn %O A037123 1,2 %A A037123 Vasiliy Danilov (danilovv@usa.net) 1998 Jun %I A062918 %S A062918 1,3,6,10,15,21,28,36,45,46,57,78,109,150,201,262,333,414,505,507,519, %T A062918 541,573,615,667,729,801,883,975,978,991,1014,1047,1090,1143,1206,1279, %U A062918 1362,1455,1459,1473,1497,1531,1575,1629,1693,1767,1851,1945,1950,1965 %N A062918 Sum of the digit reversals of first n natural numbers. %e A062918 a(12) = 78 as 1+2+3+4+5+6=7+8+9+ R(10)+R(11)+R(12) = 1+2+3+4+5+6=7+8+9+ 01+11+21 = 78. %K A062918 nonn,base,easy %O A062918 1,2 %A A062918 Amarnath Murthy (amarnath_murthy@yahoo.com), Jul 02 2001 %E A062918 More terms from Larry Reeves (larryr@acm.org), Jul 05 2001 %I A033442 %S A033442 0,0,1,3,6,10,15,21,28,36,45,54,64,75,87,100,114,129,145, %T A033442 162,180,198,217,237,258,280,303,327,352,378,405,432,460, %U A033442 489,519,550,582,615,649,684,720,756,793,831,870,910,951 %N A033442 Number of edges in 10-partite Turan graph of order n. %D A033442 Graham et al., Handbook of Combinatorics, Vol 2, p. 1234. %K A033442 nonn %O A033442 0,4 %A A033442 njas %I A033443 %S A033443 0,0,1,3,6,10,15,21,28,36,45,55,65,76,88,101,115,130,146, %T A033443 163,181,200,220,240,261,283,306,330,355,381,408,436,465, %U A033443 495,525,556,588,621,655,690,726,763,801,840,880,920,961 %N A033443 Number of edges in 11-partite Turan graph of order n. %D A033443 Graham et al., Handbook of Combinatorics, Vol 2, p. 1234. %K A033443 nonn %O A033443 0,4 %A A033443 njas %I A033444 %S A033444 0,0,1,3,6,10,15,21,28,36,45,55,66,77,89,102,116,131,147, %T A033444 164,182,201,221,242,264,286,309,333,358,384,411,439,468, %U A033444 498,529,561,594,627,661,696,732,769,807,846,886,927,969 %N A033444 Number of edges in 12-partite Turan graph of order n. %D A033444 Graham et al., Handbook of Combinatorics, Vol 2, p. 1234. %K A033444 nonn %O A033444 0,4 %A A033444 njas %I A061791 %S A061791 1,3,6,10,15,21,28,36,45,55,66,77,90,104,119,134,151,169,188,208,229, %T A061791 251,274,297,322,348,374,402,431,461,492,523,556,588,623,658,695,733, %U A061791 771,810,851,893,936,980,1025,1071,1118,1164,1213,1263,1313,1365,1417 %N A061791 Number of distinct sums i^3 + j^3 for 1<=i<=j<=n. %e A061791 If the {s+t} sums are generated by addition 2 terms of an S set consisting of n different entries, then at least 1, at most n(n+1)/2=A000217(n) distinct values can be obtained. The set of first n cubes gives results falling between these two extremes. E.g. S={1,8,27,...,2744,3375} provides 119 different sums of two, not necessarily different cubes:{2,9,....,6750}. Only a single sum occurs more then once: 1729(Ramanujan): 1729=1+1728=729+1000. %t A061791 f[x_] :=x^3 t=Table[Length[Union[Flatten[Table[f[u]+f[w],{w,1,m},{u,1,m}]]]],{m,1,75}] %Y A061791 A000217. %K A061791 nonn %O A061791 1,2 %A A061791 Labos E. (labos@ana1.sote.hu), Jun 22 2001 %I A000217 M2535 N1002 %S A000217 0,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190, %T A000217 210,231,253,276,300,325,351,378,406,435,465,496,528,561,595,630, %U A000217 666,703,741,780,820,861,903,946,990,1035,1081,1128,1176,1225,1275 %N A000217 Triangular numbers: C(n+1,2) = n(n+1)/2. %C A000217 a(n) = 0+1+2+...+n. %C A000217 Number of edges in complete graph of order n, K_n. %C A000217 For n >= 1 a(n)=n(n+1)/2 is also the genus of a nonsingular curve of degree n+2 like the Fermat curve x^(n+2) + y^(n+2) = 1 - Ahmed Fares (ahmedfares@my_deja.com), Feb 21 2001 %C A000217 a(n) is the number of ways in which (n+2) can be written as a sum of three positive integers if representations differing in the order of the terms are considered to be different. In other words a(n) is the number of positive integral solutions of the equation x + y + z = n+2. - Amarnath Murthy (amarnath_murthy@yahoo.com), Apr 22 2001 %D A000217 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828. %D A000217 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2. %D A000217 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. %D A000217 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155. %D A000217 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1. %D A000217 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954. %D A000217 T. Trotter, Some Identities for the Triangular Numbers, Journal of Recreational Mathematics, Spring 1973, 6(2). %H A000217 H. Bottomley, Illustration of initial terms of A000217, A002378 %H A000217 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A000217 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 253 %H A000217 N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326 %H A000217 T. Trotter, Some Identities for the Triangular Numbers %H A000217 E. W. Weisstein, Link to a section of The World of Mathematics (1). %H A000217 E. W. Weisstein, Link to a section of The World of Mathematics (2). %H A000217 E. W. Weisstein, Link to a section of The World of Mathematics (3). %H A000217 E. W. Weisstein, Link to a section of The World of Mathematics (4). %H A000217 E. W. Weisstein, Link to a section of The World of Mathematics (5). %H A000217 E. W. Weisstein, Link to a section of The World of Mathematics (6). %H A000217 Index entries for "core" sequences %F A000217 G.f.: x/(1-x)^3. %e A000217 When n=3, a(3) = 4*3/2 = 6. %p A000217 A000217:=proc(n) n*(n+1)/2; end; [ seq(n*(n+1)/2, n=0..100)]; %o A000217 (PARI) A000217(n)=n*(n+1)/2. %Y A000217 Cf. A007318, A002024, A000096, A000124, A002378, A000292. A000330. %Y A000217 a(2k-1)= A000384(k), a(2k)=A014105(k), k >= 1. %Y A000217 A diagonal of A008291. %K A000217 nonn,core,easy,nice %O A000217 0,3 %A A000217 njas %I A025747 %S A025747 1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231,254, %T A025747 278,303,329,356,384,413,443,474,506,539,573,608,644,681,719,758,798,839, %U A025747 881,924,969,1015,1062,1110,1159,1209,1260,1312,1365,1419,1474,1530,1587 %N A025747 Index of 10^n within sequence of numbers of form 9^i*10^j. %K A025747 nonn %O A025747 1,2 %A A025747 dww %I A025738 %S A025738 1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,191,212,234,257, %T A025738 281,306,332,359,387,416,446,477,509,542,576,611,647,684,723,763,804,846, %U A025738 889,933,978,1024,1071,1119,1168,1218,1269,1321,1374,1428,1483,1540,1598 %N A025738 Index of 9^n within sequence of numbers of form 8^i*9^j. %K A025738 nonn %O A025738 1,2 %A A025738 dww %I A025731 %S A025731 1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,137,155,174,194,215,237,260, %T A025731 284,309,335,362,390,419,449,480,513,547,582,618,655,693,732,772,813,855, %U A025731 898,942,987,1033,1081,1130,1180,1231,1283,1336,1390,1445,1501,1558,1616 %N A025731 Index of 8^n within sequence of numbers of form 7^i*8^j. %K A025731 nonn %O A025731 1,2 %A A025731 dww %I A025724 %S A025724 1,3,6,10,15,21,28,36,45,55,66,78,92,107,123,140,158,177,197,218,240,263, %T A025724 287,312,339,367,396,426,457,489,522,556,591,627,664,703,743,784,826,869, %U A025724 913,958,1004,1051,1099,1148,1198,1250,1303,1357,1412,1468,1525,1583 %N A025724 Index of 7^n within sequence of numbers of form 6^i*7^j. %K A025724 nonn %O A025724 1,2 %A A025724 dww %I A025746 %S A025746 1,3,6,10,15,21,28,36,45,55,67,80,94,109,125,142,160,179,199,221,244,268, %T A025746 293,319,346,374,403,433,465,498,532,567,603,640,678,717,757,798,841,885, %U A025746 930,976,1023,1071,1120,1170,1221,1274,1328,1383,1439,1496,1554,1613 %N A025746 Index of 10^n within sequence of numbers of form 8^i*10^j. %K A025746 nonn %O A025746 1,2 %A A025746 dww %I A025715 %S A025715 1,3,6,10,15,21,28,36,45,56,68,81,95,110,126,143,161,180,201,223,246,270, %T A025715 295,321,348,376,405,436,468,501,535,570,606,643,681,720,761,803,846,890, %U A025715 935,981,1028,1076,1125,1176,1228,1281,1335,1390,1446,1503,1561,1621 %N A025715 Index of 6^n within sequence of numbers of form 5^i*6^j. %K A025715 nonn %O A025715 1,2 %A A025715 dww %I A046489 %S A046489 1,3,6,10,15,21,28,36,45,56,78,111,155,210,276,353,441,540,641,752,873, %T A046489 1004,1145,1296,1457,1628,1809,2000,2202,2414,2636,2868,3110,3362,3624, %U A046489 3896,4178,4470,4773,5086,5409,5742,6085,6438,6801,7174,7557,7950,8354 %N A046489 Sum of the first n palindromes. %H A046489 P. De Geest, World!Of Numbers %Y A046489 Cf. A002113, A007504, A046485. %K A046489 nonn %O A046489 0,2 %A A046489 Patrick De Geest (pdg@worldofnumbers.com), Sep 1998. %I A050760 %S A050760 0,1,3,6,10,15,21,28,36,45,78,91,105,120,136,153,171,190,210,231,253, %T A050760 276,325,351,378,406,435,465,496,528,561,595,630,703,741,780,820,861, %U A050760 903,946,1035,1081,1275,1326,1378,1431,1485,1540,1596,1653,1830,1891 %N A050760 Decimal expansion of triangular number k (values of k see A050759) contains no pair of consecutive equal digits. %Y A050760 Cf. A043096, A000217, A050759. %K A050760 nonn,base %O A050760 0,3 %A A050760 Patrick De Geest (pdg@worldofnumbers.com), Sep 1999. %I A025737 %S A025737 1,3,6,10,15,21,28,36,46,57,69,82,96,111,127,144,163,183,204,226,249,273, %T A025737 298,324,352,381,411,442,474,507,541,577,614,652,691,731,772,814,857,902, %U A025737 948,995,1043,1092,1142,1193,1245,1299,1354,1410,1467,1525,1584,1644 %N A025737 Index of 9^n within sequence of numbers of form 7^i*9^j. %K A025737 nonn %O A025737 1,2 %A A025737 dww %I A025706 %S A025706 1,3,6,10,15,21,28,37,47,58,70,83,97,113,130,148,167,187,208,231,255,280, %T A025706 306,333,361,391,422,454,487,521,556,592,630,669,709,750,792,835,880,926, %U A025706 973,1021,1070,1120,1172,1225,1279,1334,1390,1447,1506,1566,1627,1689 %N A025706 Index of 5^n within sequence of numbers of form 4^i*5^j. %Y A025706 Differs from A025730 at a(56). %K A025706 nonn %O A025706 1,2 %A A025706 dww %I A025730 %S A025730 1,3,6,10,15,21,28,37,47,58,70,83,97,113,130,148,167,187,208,231,255,280, %T A025730 306,333,361,391,422,454,487,521,556,592,630,669,709,750,792,835,880,926, %U A025730 973,1021,1070,1120,1172,1225,1279,1334,1390,1447,1506,1566,1627,1689 %N A025730 Index of 8^n within sequence of numbers of form 6^i*8^j. %Y A025730 Differs from A025706 at a(56). %K A025730 nonn %O A025730 1,2 %A A025730 dww %I A062099 %S A062099 0,1,3,6,10,15,21,28,55,78,91,105,120,136,190,210,231,253,276,300,325, %T A062099 406,465,528,703,780,820,861,1081,1176,1225,1275,1540,1596,1653,1711, %U A062099 1770,2080,2211,2346,2701,2775,2850,3003,3160,3403,3486,3570,3741,3828 %N A062099 Triangular numbers with the sum of the digits a triangular number. %e A062099 a(8) = 28 is a triangular number, and the sum of digits 10 is also a triangular number. %Y A062099 A000217. %K A062099 nonn,base,easy %O A062099 0,3 %A A062099 Amarnath Murthy (amarnath_murthy@yahoo.com), Jun 16 2001 %E A062099 More terms from Erich Friedman (efriedma@stetson.edu), Jun 20 2001 %I A051166 %S A051166 1,0,0,1,3,6,10,15,21,29 %V A051166 1,0,0,-1,3,-6,10,-15,21,-29 %N A051166 Binomial transform of (a0,a1,a2,a3,...) = (a0,a0,a0,a1,a1,a1,a2,a2,a2,a3,a3,a3,...). %H A051166 N. J. A. Sloane, Transforms %Y A051166 Cf. A051163, A051164, A051165. %K A051166 easy,sign,more,eigen %O A051166 0,5 %A A051166 Jonas Wallgren (jonwa@ida.liu.se) %I A025745 %S A025745 1,3,6,10,15,21,29,38,48,59,71,85,100,116,133,151,170,191,213,236,260, %T A025745 285,312,340,369,399,430,462,496,531,567,604,642,682,723,765,808,852,897, %U A025745 944,992,1041,1091,1142,1195,1249,1304,1360,1417,1475,1535,1596,1658 %N A025745 Index of 10^n within sequence of numbers of form 7^i*10^j. %K A025745 nonn %O A025745 1,2 %A A025745 dww %I A061304 %S A061304 1,3,6,10,15,21,55,66,78,91,105,190,210,231,253,406,435,465,561,595,703, %T A061304 741,861,903,946,1081,1326,1378,1653,1711,1770,1830,1891,2145,2211,2278, %U A061304 2346,2415,2485,2701,2926,3003,3081,3403,3486,3570,3655,3741,4186,4278 %N A061304 Square-free triangular numbers. %e A061304 105 = 3 * 5 * 7 is a square-free triangular number. %Y A061304 Cf. A000217. %K A061304 nonn,easy %O A061304 0,2 %A A061304 Amarnath Murthy (amarnath_murthy@yahoo.com), Apr 26 2001 %E A061304 More terms from Asher Natan Auel (auela@reed.edu), May 14, 2001 %I A025723 %S A025723 1,3,6,10,15,22,30,39,49,60,73,87,102,118,135,154,174,195,217,240,265, %T A025723 291,318,346,376,407,439,472,506,542,579,617,656,696,738,781,825,870,916, %U A025723 964,1013,1063,1114,1166,1220,1275,1331,1388,1447,1507,1568,1630,1693 %N A025723 Index of 7^n within sequence of numbers of form 5^i*7^j. %K A025723 nonn %O A025723 1,2 %A A025723 dww %I A022784 %S A022784 1,3,6,10,15,22,30,39,49,61,74,88,103,119,137,156,176,197,220,244, %T A022784 269,295,322,351,381,412,444,478,513,549,586,624,664,705,747,790, %U A022784 835,881,928,976,1025,1076,1128,1181,1235,1291,1348,1406,1465,1526 %N A022784 Place where n-th 1 occurs in A023122. %K A022784 nonn %O A022784 1,2 %A A022784 Clark Kimberling (ck6@cedar.evansville.edu) %I A025736 %S A025736 1,3,6,10,15,22,30,39,49,61,74,88,103,119,137,156,176,197,220,244,269, %T A025736 295,322,351,381,412,444,478,513,549,586,625,665,706,748,791,836,882,929, %U A025736 977,1027,1078,1130,1183,1237,1293,1350,1408,1467,1528,1590,1653,1717 %N A025736 Index of 9^n within sequence of numbers of form 6^i*9^j. %K A025736 nonn %O A025736 1,2 %A A025736 dww %I A022952 %S A022952 3,6,10,15,22,30,39,50,62,75,89,105,122,140,159,179,200,223,247,272, %T A022952 298,325,353,382,413,445,478,512,547,583,620,658,698,739,781,824, %U A022952 868,913,959,1006,1054,1103,1154,1206,1259,1313,1368,1424,1481,1539 %N A022952 a(n) = a(n-1) + c(n) for n >= 3, a( ) increasing, given a(1)=3 a(2)=6; where c( ) is complement of a( ). %K A022952 nonn %O A022952 1,1 %A A022952 Clark Kimberling (ck6@cedar.evansville.edu) %I A024918 %S A024918 1,3,6,10,15,22,30,39,50,63,79,96,115,138,163,190,219,250,282,319, %T A024918 360,403,450,499,552,611,672,736,803,874,947,1026,1107,1190,1279, %U A024918 1376,1477,1580,1687,1796,1909,2030,2155,2282,2410,2541,2678,2817 %N A024918 Partial sums of the sequence of prime powers (A000961). %Y A024918 Cf. A000961, A024923. %K A024918 nonn %O A024918 0,2 %A A024918 Den Roussel (DenRoussel@webtv.net) %I A011914 %S A011914 0,0,0,0,0,1,3,6,10,15,22,30,41,53,68,85,105,127,153,181, %T A011914 213,249,288,332,379,431,487,548,614,685,761,842,930,1023, %U A011914 1122,1227,1338,1456,1581,1713,1852,1998,2152,2313,2483 %N A011914 [ n(n-1)(n-2)/32 ]. %K A011914 nonn %O A011914 0,7 %A A011914 njas %I A026104 %S A026104 1,3,6,10,15,22,33,44,55,68,85,102,119,145,174,203,232,261,296,333,370,410,451 %N A026104 a(n) = greatest number in row n of A026098 that is not a positive power of 2. %K A026104 nonn %O A026104 1,2 %A A026104 Clark Kimberling (ck6@cedar.evansville.edu) %I A063542 %S A063542 0,1,3,6,10,15,23 %N A063542 Least number of empty convex 4-gons determined by n points in the plane. %H A063542 O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001. %Y A063542 Cf. A063541. %K A063542 nonn %O A063542 4,3 %A A063542 njas, Aug 14 2001 %I A024674 %S A024674 1,3,6,10,15,24,33,42,52,67,80,95,112,131,150,170,195,220,243,268,300, %T A024674 328,357,388,425,459,492,528,571,612,652,692,743,788,833,879,937,985, %U A024674 1035,1092,1150,1206,1262,1324,1388,1448,1507,1579,1647,1712,1777,1857 %N A024674 a(n) = position of n^3 + (n+1)^3 in A024670 (distinct sums of cubes of distinct positive integers). %K A024674 nonn %O A024674 1,2 %A A024674 Clark Kimberling (ck6@cedar.evansville.edu) %I A058576 %S A058576 1,3,6,10,15,24,37 %N A058576 McKay-Thompson series of class 24F for Monster. %D A058576 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994). %Y A058576 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc. %K A058576 nonn %O A058576 -1,2 %A A058576 njas, Nov 27, 2000 %I A025215 %S A025215 1,3,6,10,16,22,28,36,44,52,61,71,81,92,103,114,126,138,151,164,177,191,205, %T A025215 219,233,248,263,279,295,311,327,343,360,377,394,412,429,447,465,483,502, %U A025215 521,540,559,578,598,617,637,657,677,698,718,739,760,781,803,824,846,867 %N A025215 a(n) = [ Sum of squares of 1 + 1/2 + ... + 1/k ], k = 1,2,...,n. %K A025215 nonn %O A025215 1,2 %A A025215 Clark Kimberling (ck6@cedar.evansville.edu) %I A049697 %S A049697 1,3,6,10,16,22,30,40,50,60,74,88,104,122,136,152,176,198,222,248,268, %T A049697 290,322,352,380,412,442,472,512,548,586,632,668,704,744,780,828,882, %U A049697 924,964,1020,1072,1126 %N A049697 a(n)=T(n,n+1), array T as in A049695. %K A049697 nonn %O A049697 0,2 %A A049697 Clark Kimberling, ck6@cedar.evansville.edu %I A025701 %S A025701 1,3,6,10,16,23,31,40,51,63,76,90,106,123,141,160,181,203,226,250,276, %T A025701 303,331,361,392,424,457,492,528,565,603,643,684,726,769,814,860,907,955, %U A025701 1005,1056,1108,1161,1216,1272,1329,1388,1448,1509,1571,1635,1700,1766 %N A025701 Index of 4^n within sequence of numbers of form 3^i*4^j. %K A025701 nonn %O A025701 1,2 %A A025701 dww %I A025744 %S A025744 1,3,6,10,16,23,31,40,51,63,76,91,107,124,142,162,183,205,229,254,280, %T A025744 307,336,366,397,430,464,499,535,573,612,652,694,737,781,826,873,921,970, %U A025744 1021,1073,1126,1180,1236,1293,1351,1411,1472,1534,1597,1662,1728,1795 %N A025744 Index of 10^n within sequence of numbers of form 6^i*10^j. %K A025744 nonn %O A025744 1,2 %A A025744 dww %I A025714 %S A025714 1,3,6,10,16,23,31,41,52,64,77,92,108,125,144,164,185,207,231,256,282, %T A025714 310,339,369,401,434,468,503,540,578,617,658,700,743,787,833,880,928,978, %U A025714 1029,1081,1134,1189,1245,1302,1361,1421,1482,1545,1609,1674,1740,1808 %N A025714 Index of 6^n within sequence of numbers of form 4^i*6^j. %Y A025714 Differs from A025729 at a(65). %K A025714 nonn %O A025714 1,2 %A A025714 dww %I A025729 %S A025729 1,3,6,10,16,23,31,41,52,64,77,92,108,125,144,164,185,207,231,256,282, %T A025729 310,339,369,401,434,468,503,540,578,617,658,700,743,787,833,880,928,978, %U A025729 1029,1081,1134,1189,1245,1302,1361,1421,1482,1545,1609,1674,1740,1808 %N A025729 Index of 8^n within sequence of numbers of form 5^i*8^j. %Y A025729 Differs from A025714 at a(65). %K A025729 nonn %O A025729 1,2 %A A025729 dww %I A011913 %S A011913 0,0,0,0,0,1,3,6,10,16,23,31,42,55,70,88,108,131,157,187, %T A011913 220,257,298,342,391,445,503,566,634,707,785,870,960,1056, %U A011913 1158,1266,1381,1503,1632,1768,1912,2063,2221,2388,2563 %N A011913 [ n(n-1)(n-2)/31 ]. %K A011913 nonn %O A011913 0,7 %A A011913 njas %I A024531 %S A024531 0,1,3,6,10,16,23,32,42,55,70,87,106,126,148,173,201,230,262,296,331,369, %T A024531 409,452,499,548,598,650,703,758,820,883,950,1018,1091,1164,1241,1321,1402, %U A024531 1487,1575,1664,1757,1852 %N A024531 a(n) = [ (2nd elementary symmetric function of P(n))/(1st elementary symmetric function of P(n)) ], where P(n) = {1, p(1), p(2), ..., p(n-1)}, where p(0) = 1. %K A024531 nonn %O A024531 2,3 %A A024531 Clark Kimberling (ck6@cedar.evansville.edu) %I A034198 %S A034198 0,1,3,6,10,16,23,32,43,56,71,89,109,132 %N A034198 Binary codes (not necessarily linear) of length n with 3 words. %D A034198 H. Fripertinger, Enumeration, construction and random generation of block codes, Designs, Codes, Crypt., 14 (1998), 213-219. %H A034198 H. Fripertinger, Isometry Classes of Codes %Y A034198 Cf. A034188-. %K A034198 nonn %O A034198 1,3 %A A034198 njas %I A025222 %S A025222 0,0,1,3,6,10,16,25,35,48,65,84,108,135,166,203,244,290,342,400,464,535,613, %T A025222 698,790,999,1116,1242,1377,1522,1677,1842,2017,2203,2401,2610,2831,3064,3309, %U A025222 3568,3839,4125,4423,4737,5064,5407,5764,6138,6527,6932,7353,7792,8248 %N A025222 a(n) = [ Sum{(sqrt(j+1)-sqrt(i+1))^2} ], 1 <= i < j <= n. %K A025222 nonn %O A025222 1,4 %A A025222 Clark Kimberling (ck6@cedar.evansville.edu) %I A011902 %S A011902 0,0,0,0,1,3,6,10,16,25,36,49,66,85,109,136,168,204,244, %T A011902 290,342,399,462,531,607,690,780,877,982,1096,1218,1348, %U A011902 1488,1636,1795,1963,2142,2331,2530,2741,2964,3198,3444 %N A011902 [ n(n-1)(n-2)/20 ]. %K A011902 nonn %O A011902 0,6 %A A011902 njas %I A025004 %S A025004 3,6,10,16,25,36,51,70,94,124,161,207,262,328,407,502,614,746,900,1080, %T A025004 1288,1529,1808,2127,2494,2915,3393,3939,4556,5253,6040,6930,7931,9056, %U A025004 10322,11729,13308,15067,17031,19208,21637,24340,27330,30633,34296,38344 %N A025004 a(1) = 3; a(n+1) = a(n)-th nonprime, where nonprimes begin at 1. %K A025004 nonn %O A025004 1,1 %A A025004 dww %I A054886 %S A054886 1,3,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,8362, %T A054886 13530,21892,35422,57314,92736,150050,242786,392836,635622,1028458, %U A054886 1664080,2692538,4356618,7049156,11405774,18454930,29860704,48315634 %N A054886 Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3,Pi/3,0) (this is the classical modular tessellation). %C A054886 The layer sequence is the sequence of the cardinalities of the layers accumulating around a ( finite-sided ) polygon of the tessellation under successive side-reflections; see the illustration accompanying A054888. %C A054886 Also spherical growth series for modular group. %D A054886 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 156. %H A054886 Index entries for sequences related to modular groups %F A054886 G.f.: (1+2*x+2*x^2+x^3)/(1-x-x^2) = (x^2+x+1)*(1+x)/(1-x-x^2). a(n)=2*F(n) for n>2, with F(n) the n-th Fibonacci number (cf. A000045 ) %K A054886 nonn,easy,nice %O A054886 1,2 %A A054886 Paolo Dominici (pd@full-service.it), May 23 2000 %I A038505 %S A038505 0,1,3,6,10,16,28,56,120,256,528,1056,2080,4096,8128,16256,32640,65536, %T A038505 131328,262656,524800,1048576,2096128,4192256,8386560,16777216, %U A038505 33558528,67117056,134225920,268435456,536854528,1073709056 %N A038505 Sum of every 4th entry of row n in Pascal's triangle, starting at "n choose 2". %K A038505 easy,nonn %O A038505 0,3 %A A038505 Frank Ruskey (fruskey@csr.csc.UVic.CA) %I A005045 M2536 %S A005045 1,3,6,10,17,25,37,51,70,92,121,153,194,240,296,358,433,515,612,718, %T A005045 841,975,1129,1295,1484,1688,1917,2163,2438,2732,3058,3406,3789,4197,4644 %N A005045 3 X 3 matrices with row and column sums n. %D A005045 E. J. Morgan, Construction of Block Designs and Related Results. Ph.D. Dissertation, Univ. Queensland, 1978. %D A005045 E. J. Morgan, On 3 X 3 matrices with constant row and column sum, Abstract 763-05-13, Notices Amer. Math. Soc., 26 (1979), page A-27. %K A005045 nonn,nice %O A005045 2,2 %A A005045 njas %I A029864 %S A029864 1,3,6,10,18,33,50,85,135,206,319,488,714,1068,1559,2241, %T A029864 3226,4598,6448,9076,12622,17415,23982,32797,44496,60311, %U A029864 81171,108698,145178,192947,255189,336804,442434,579093 %N A029864 Euler transform of 1 2 3 1 2 3 1 2 3 .... %H A029864 N. J. A. Sloane, Transforms %K A029864 nonn %O A029864 0,2 %A A029864 njas %I A017991 %S A017991 1,1,3,6,10,19,36,65,118,216,392,713,1296,2354,4279,7776, %T A017991 14129,25675,46656,84779,154054,279936,508677,924328,1679616, %U A017991 3052064,5545969,10077696,18312388,33275819,60466176,109874333 %N A017991 Powers of cube root of 6 rounded down. %K A017991 nonn %O A017991 0,3 %A A017991 njas %I A058356 %S A058356 1,1,3,6,10,19,37,74,140,269,520,1026,1984,3831,7368,14302,27707,53729, %T A058356 103826,201035,389094,753858,1458852,2824336,5466167,10584567,20489682, %U A058356 39669401,76787404,148660956,287786132,557153147,1078562051,2088027468 %N A058356 Coefficients in the series (1 + 2x^2 + 3x^3 + 5x^5 + 7x^7 + 11x^11 + 13x^13 + ... )/(1 - 4x^4 - 6x^6 - 8x^8 - 9x^9 - 10x^10 - 12x^12 - 14x^14 - ... ). %t A058356 CoefficientList[ Series[ (1 + Apply[ Plus, Select[ Range[ 2, 50 ], PrimeQ[ # ] & ]*x^Select[ Range[ 2, 50 ], PrimeQ[ # ] & ] ] ) / (1 - Apply[ Plus, Select[ Range[ 1, 50 ], !PrimeQ[ # ] & ]*x^Select[ Range[ 1, 50 ], !PrimeQ[ # ] & ] ] ), {x, 0, 40} ], x ] %K A058356 nonn %O A058356 0,3 %A A058356 Robert G. Wilson v (rgwv@kspaint.com), Dec 16 2000 %I A018171 %S A018171 1,1,3,6,10,20,36,66,120,219,400,728,1325,2413,4394,8000, %T A018171 14564,26515,48273,87884,160000,291290,530312,965468,1757696, %U A018171 3200000,5825805,10606252,19309364,35153937,64000000,116516108 %N A018171 Powers of fifth root of 20 rounded down. %K A018171 nonn %O A018171 0,3 %A A018171 njas %I A060179 %S A060179 1,3,6,10,21,21,50,73,116,167,248,385,496,728,959,1548,1899,2835,3609, %T A060179 5042 %N A060179 Sum of distinct orders of degree n permutations. %e A060179 Set of orders of all degree 7 permutations is {1,2,3,4,5,6,7,10,12) so a(7)=1+2+3+4+5+6+7+10+12=50. %Y A060179 Cf. A060014, A060015. %K A060179 easy,more,nonn %O A060179 1,2 %A A060179 Vladeta Jovovic (vladeta@Eunet.yu), Mar 19 2001 %I A056411 %S A056411 3,6,10,21,24,92,78,327,443,1632,1698,12769,10464,57840,122822,348222, %T A056411 476052,3597442,3401970,22006959,41597374,142677588,186077886, %U A056411 1476697627,1694658003,8147282460,15690973754 %N A056411 Step cyclic shifted sequences using a maximum of three different characters. %C A056411 See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent. %D A056411 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. %D A056411 R. C. Titsworth, (1964). Equivalence classes of periodic sequences. Illinois J. Math. 8: 266-270. %F A056411 Refer to Titsworth or slight "simplification" in Nester. %Y A056411 Cf. A002729. %K A056411 nonn %O A056411 1,1 %A A056411 Marks R. Nester (nesterm@qfri1.se2.dpi.qld.gov.au) %I A061883 %S A061883 1,1,1,3,6,10,21,36,78,153,300,595,1176,2346,4656,9316,18528,37128, %T A061883 74305,148240,296835,593505,1186570,2372931,4744740,9489546,18975880, %U A061883 37953828,75909681,151806600,303626403,607243825,1214480970,2428940451 %N A061883 Largest triangular number less than or equal to sum of previous terms with a(0)=1. %C A061883 a(5)=10 since 1+1+1+3+6=12 and 10 is the largest triangular number less than or equal to this. %F A061883 a(n) = A060985(n+1)-A060985(n) = A057944(A060985(n)) = A000217(A003056(A060985(n))) %K A061883 nonn %O A061883 0,4 %A A061883 Henry Bottomley (se16@btinternet.com), May 12 2001 %I A027671 %S A027671 1,3,6,10,21,39,92,198,498,1219,3210,8418,22913,62415,173088, %T A027671 481598,1351983,3808083,10781954,30615354,87230157,249144711, %U A027671 713387076,2046856566,5884491500,16946569371,48883660146 %N A027671 Necklaces with n beads of 3 colors, allowing turning over. %D A027671 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. %D A027671 M. Gardner "New Mathematical Diversions from Scientific American" (Simon and Schuster, New York, 1966), pages 245-246. %D A027671 J. L. Fisher, Application-Oriented Algebra (1977) ISBN 0-7002-2504-8, circa p 215. %H A027671 Index entries for sequences related to bracelets %H A027671 F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. %H A027671 E. W. Weisstein, Link to a section of The World of Mathematics. %K A027671 nonn,easy,nice %O A027671 0,2 %A A027671 Alford Arnold (Alford1940@aol.com) %E A027671 More terms from Christian G. Bower (bowerc@usa.net) %I A005522 M2537 %S A005522 3,6,10,21,46,108,263,658,1674,4305,11146,28980 %N A005522 From sum of 1/F(n), where F() = Fibonacci numbers A000045. %D A005522 W. E. Greig, Sums of Fibonacci reciprocals, Fib. Quart., 15 (1977), 46-48. %K A005522 nonn %O A005522 1,1 %A A005522 njas %I A063015 %S A063015 1,3,6,10,22,30,42,46,58,82,102,106,110,114,138,166,174,178,182,210, %T A063015 226,230,258,262,282,318,330,346,354,358,374,382,402,410,434,462,466, %U A063015 478,502,546,562,570,586,602,618,642,654,678,690,710,718,762,822,830 %N A063015 n + mu(n) is prime. %t A063015 Select[ Range[ 1,830 ],PrimeQ[ #+MoebiusMu[ # ] ]& ] %Y A063015 Cf. A063452. %K A063015 nonn %O A063015 0,2 %A A063015 Dean Hickerson (dean@math.ucdavis.edu) %I A048006 %S A048006 0,0,0,3,6,10,25,45,77,175,322,570,1245,2325,4213,9031,17061,31421, %T A048006 66547,126763,236203,496063,950818,1787346,3730293,7184421,13598053, %U A048006 28243063,54604081,103918153,215008363,416990563,797154723 %N A048006 Number of nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= (n-1)/3. %K A048006 nonn %O A048006 1,4 %A A048006 Clark Kimberling, ck6@cedar.evansville.edu %I A001465 M2538 N1003 %S A001465 0,0,1,3,6,10,30,126,448,1296,4140,17380,76296,296088,1126216,4940040, %T A001465 23904000,110455936,489602448,2313783216,11960299360,61878663840 %N A001465 Degree n odd permutations of order 2. %D A001465 L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168. %K A001465 nonn %O A001465 0,4 %A A001465 njas,jhc %I A062100 %S A062100 0,1,3,6,10,36,66,136,300,630,666,3003,3160,10011,13366,16110,60031, %T A062100 66066,106030,163306,303031,333336,336610,600060,630003,1010331, %U A062100 1030330,1063611,1313010,3306306,3316600,6601161,10006101,11066160 %N A062100 Triangular numbers with the every digit a triangular number. %e A062100 a(7) = 36 is a triangular number, 3 and 6 are also triangular numbers. %Y A062100 A000217. %K A062100 nonn,base,easy %O A062100 0,3 %A A062100 Amarnath Murthy (amarnath_murthy@yahoo.com), Jun 16 2001 %E A062100 Corrected and extended by Larry Reeves (larryr@acm.org), Jun 22 2001 %I A048089 %S A048089 0,0,0,0,0,3,6,10,40,60,84,210,280,360,846,1110,1430,4235,5775,7735, %T A048089 24388,33124,44100,128485,170968,223720,639880,846600,1105800,3311187, %U A048089 4401540,5784240,17717910,23544510,30936334,93818725 %N A048089 Number of nonempty subsets of {1,2,...,n} in which exactly 3/5 of the elements are <= (n+3)/3. %K A048089 nonn %O A048089 1,6 %A A048089 Clark Kimberling, ck6@cedar.evansville.edu %I A061455 %S A061455 1,3,6,10,55,66,120,153,171,190,300,351,595,630,666,820,3003,5995,8778, %T A061455 15051,17578,66066,87571,156520,180300,185745,547581,557040,617716, %U A061455 678030,828828,1269621,1461195,1680861,1851850,3544453,5073705,5676765 %N A061455 Triangular numbers whose digit reversal is also a triangular number. %e A061455 153 and 351 both are triangular numbers. %Y A061455 Cf. A000217. %K A061455 nonn,base,easy %O A061455 0,2 %A A061455 Amarnath Murthy (amarnath_murthy@yahoo.com), May 03 2001 %E A061455 More terms from Erich Friedman (efriedma@stetson.edu), May 08 2001 %I A061380 %S A061380 0,1,3,6,10,66,105,120,153,190,210,231,300,351,406,465,630,703,741,780, %T A061380 820,903,990,1035,1081,1326,1540,1770,1830,2016,2080,2556,2701,2850, %U A061380 3003,3081,3160,3240,3403,3570,4005,4095,4560,4950,5050,5460,5671,6105 %N A061380 Triangular numbers with product of digits also a triangular number. %e A061380 153 is a triangular number and the product of digits 15 is also a triangular number. %K A061380 nonn,base,easy %O A061380 0,3 %A A061380 Amarnath Murthy (amarnath_murthy@yahoo.com), May 02 2001 %E A061380 More terms from Erich Friedman (efriedma@stetson.edu), May 08 2001 %I A006509 M2539 %S A006509 1,3,6,11,4,15,2,19,38,61,32,63,26,67,24,71,18,77,16,83,12,85,164,81, %T A006509 170,73,174,277,384,275,162,35,166,29,168,317,468,311,148,315,142,321, %U A006509 140,331,138,335,136,347,124,351,122,355,116,357,106,363,100,369,98 %N A006509 Cald's sequence: a(n+1)=a(n)-p(n) if new and >0, else a(n)+p(n) if new, otherwise 0, where p are primes. %D A006509 F. Cald, Problem 356, Franciscan order, J. Rec. Math., 7 (No. 4, 1974), 318; 10 (No. 1, 1974), 62-64. %D A006509 "Cald's Sequence", Popular Computing (Calabasas, CA), Vol. 4 (No. 41, Aug 1976), pp. 16-17. %K A006509 nonn,nice,easy %O A006509 1,2 %A A006509 njas %E A006509 More terms from Larry Reeves (larryr@acm.org), Jul 20 2001 %I A028744 %S A028744 3,6,11,12,13,15,17,21,22,23,24,26,27,29,30 %N A028744 Nonsquares mod 31. %K A028744 nonn,fini,full %O A028744 0,1 %A A028744 njas %I A028775 %S A028775 3,6,11,12,13,15,17,21,22,23,24,26,27,29,30,31,34,37,42, %T A028775 43,44,46,48,52,53,54,55,57,58,60,61 %N A028775 Nonsquares mod 62. %K A028775 nonn,fini,full %O A028775 0,1 %A A028775 njas %I A022155 %S A022155 3,6,11,12,13,15,19,22,24,25,26,30,35,38,43,44,45,47,48,49,50,52,53, %T A022155 55,59,60,61,63,67,70,75,76,77,79,83,86,88,89,90,94,96,97,98,100, %U A022155 101,103,104,105,106,110,115,118,120,121,122,126,131,134,139,140 %N A022155 Values of n at which Golay-Rudin-Shapiro sequence A020985 is negative. %D A022155 J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869. %K A022155 nonn %O A022155 1,1 %A A022155 Robert G. Wilson v (rgwv@kspaint.com) %I A000419 %S A000419 3,6,11,12,14,19,21,22,24,27,30,33,35,38,42,43,44,46,48,51,54,56,57,59, %T A000419 62,66,67,69,70,75,76,77,78,83,84,86,88,91,93,94,96,99,102,105,107,108, %U A000419 110,114,115,118,120,123,126,129,131,132,133,134,138,139,140,141,142 %N A000419 Sum of 3 but no fewer nonzero squares. %D A000419 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 311. %H A000419 E. W. Weisstein, Link to a section of The World of Mathematics. %H A000419 Index entries for sequences related to sums of squares %F A000419 Legendre: a nonnegative integer is a sum of three (or fewer) squares iff it is not of the form 4^k m with m == 7 (mod 8). %Y A000419 Cf. A000378, A000408, A000415, A002828, A004215. %K A000419 nonn,nice,easy %O A000419 1,1 %A A000419 njas,jhc %E A000419 More terms from Arlin Anderson (arlin@myself.com) %I A058269 %S A058269 1,3,6,11,12,23,19,34,33,46,37,73,47,74,75,98,71,127,83,144,120, %T A058269 143,111,213,137,183,173,230,157,288,173,279,232,272,237,392,226, %U A058269 320,296,419,263,463,282,443,404,426,323,610,362,525,440,566,386 %N A058269 An approximation to sigma_{3/2}(n): floor( sum_{d|n} d^(3/2) ). %p A058269 f:=proc(n) local d, t1, t2; t2 := 0; t1 := divisors(n); for d in t1 do t2 := t2 + d^(3/2) end do; t2; end proc; # exact value of sigma_{3/2}(n) %K A058269 nonn %O A058269 1,2 %A A058269 njas, Dec 08 2000 %I A063275 %S A063275 3,6,11,14,19,21,22,30,38,39,42,46,47,51,55,56,60,62,66,67,69,70,71,75, %T A063275 77,78,79,83,84,86,92,93,94,95,102,103,105,107,110,114,115,118,120,123, %U A063275 131,138,139,142,143,147,151,154,156,158,159,163,165,166,167,168,175 %N A063275 Numbers that require three powerful numbers (def. 1) to sum to them. %e A063275 The powerful numbers (A001694) start 1,4,8,9,... Now 11 = 1+1+9, and is not the sum of fewer terms, so 11 is in the sequence. %Y A063275 Cf. A001694, A056828, A063274. %K A063275 nonn %O A063275 0,1 %A A063275 Jud McCranie (jud.mccranie@mindspring.com), Jul 13 2001 %I A026368 %S A026368 3,6,11,14,19,22,25,28,33,36,41,44,47,50,55,58,63,66,71,74,79, %T A026368 82,85,88,93,96,101,104,107,110,115,118,123,126,131,134,139,142, %U A026368 145,148,153,156,161,164,167,170,175,178,183,186,189,192,197 %N A026368 a(n) = greatest k such that s(k) = n, where s = A026366. %K A026368 nonn %O A026368 1,1 %A A026368 Clark Kimberling, ck6@cedar.evansville.edu %I A047398 %S A047398 3,6,11,14,19,22,27,30,35,38,43,46,51,54,59,62,67,70,75,78,83,86,91,94, %T A047398 99,102,107,110,115,118,123,126,131,134,139,142,147,150,155,158,163, %U A047398 166,171,174,179,182,187,190 %N A047398 Congruent to {3, 6} mod 8. %K A047398 nonn %O A047398 0,1 %A A047398 njas %I A047924 %S A047924 3,6,11,14,19,24,27,32,35,40,45,48,53,58,61,66,69,74,79,82,87,90,95, %T A047924 100,103,108,113,116,121,124,129,134,137,142,147,150,155,158,163,168, %U A047924 171,176,179,184,189,192,197,202,205,210,213,218,223,226,231,234,239 %N A047924 2nd column of array in A038150. %D A047924 A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288. %H A047924 A. S. Fraenkel, Arrays, numeration systems and games. %K A047924 nonn,nice,easy %O A047924 0,1 %A A047924 njas %E A047924 More terms from Naohiro Nomoto (6284968128@geocities.co.jp), Jun 08 2001 %I A015823 %S A015823 3,6,11,14,23,24,30,33,35,54,59,66,71,77,99,102,105,107,110,119,131, %T A015823 140,158,161,167,168,174,179,186,203,222,231,239,251,253,280,287, %U A015823 290,304,311,312,329,347,348,357,359,366,371,378,383,402,413,419 %N A015823 phi(n + 7) | sigma(n). %K A015823 nonn %O A015823 0,1 %A A015823 Robert G. Wilson v (rgwv@kspaint.com) %I A049620 %S A049620 0,0,3,6,11,14,23,26,35,42,55,58,75,78,95,110,127,130,155,158,183,202, %T A049620 227,230,263,274,303,322,355,358,403,406,439,466,503,526,575,578 %N A049620 a(n)=T(n,n), array T as in A049615. %K A049620 nonn %O A049620 0,3 %A A049620 Clark Kimberling, ck6@cedar.evansville.edu %I A056232 %S A056232 3,6,11,15,18,22,25,27,30,35,39,41,44,51,59,63,66,70,73,75,81,88, %T A056232 91,94,97,99,103,107,109,111,114,118,121,123,126,131,135,137,140, %U A056232 147,153,155,157,161,165,167,170,173,176,181,188,194,198,201,203 %N A056232 Form an array with 3 rows: row 1 begins with 1; all rows are increasing; each entry is sum of 2 entries above it; each number appears at most once; smallest unused number is appended to first row if possible. Sequence gives row 2. %e A056232 Array begins %e A056232 1 2 4 7 8 10 12 ... %e A056232 .3 6 11 15 18 ... %e A056232 . 9 17 26 33 ... %Y A056232 Cf. A056231, A056233, A056234. See also A057153, A052474, A057154, A056230. %K A056232 nonn,nice,easy %O A056232 1,1 %A A056232 njas, E. M. Rains, Aug 22 2000 %I A059753 %S A059753 1,3,6,11,15,22,30,41,48,61 %N A059753 Minimal degree of a height one multiple of (x-1)^n. %D A059753 P. Borwein and M. J. Mossinghoff, Polynomials with Height 1 and Prescribed Vanishing at 1, Experimental Mathematics, 9:3 (2000), 425-433. %H A059753 Experimental Mathematics, Home Page %K A059753 nonn,nice %O A059753 1,2 %A A059753 njas, Feb 11 2001 %E A059753 The reference gives upper bounds for n = 11 ... 21, for example a(11) <= 69, a(12) <= 93, a(13) <= 112. %I A030722 %S A030722 1,3,6,11,16,20,25,36,44,45,53,63,75,88,104,120,140,163,188,216, %T A030722 247,280,316,354,395,438,483,530,580,633,689 %N A030722 n-th number k such that s(k)>s(j) for j=1,2,...,k-1, where s=A030717 (with s(0)=0). %K A030722 nonn %O A030722 1,2 %A A030722 Clark Kimberling, ck6@cedar.evansville.edu %I A030720 %S A030720 1,3,6,11,16,20,25,36,44,52,45,74,192,73,53,103,86,63,121,75, %T A030720 636 %N A030720 a(n)=least k such that s(k)=n, where s=A030717. %K A030720 nonn %O A030720 1,2 %A A030720 Clark Kimberling, ck6@cedar.evansville.edu %I A024401 %S A024401 1,3,6,11,16,22,30,38,47,58,69,81,95,109,124,141,158,176,196,216,237,260, %T A024401 283,307,333,359,386,415,444,474,506,538,571,606,641,677,715,753,792,833, %U A024401 874,916,960 %N A024401 a(n) = [ (3rd elementary symmetric function of S(n))/(2nd elementary symmetric function of S(n)) ], where S(n) = {first n+2 positive integers congruent to 2 mod 3}. %K A024401 nonn %O A024401 1,2 %A A024401 Clark Kimberling (ck6@cedar.evansville.edu) %I A034029 %S A034029 1,3,6,11,17,18,27,33,34,38,43,51,57,66,67,81,82,83,86,89,99,102, %T A034029 113,114,118,121,123,129,131,139,146,153,162,171,177,179,187,193, %U A034029 194,198,201,209,214,219,227,233,241,242,246,249,257,258,262,267 %N A034029 Primitively represented by x^2+2y^2. %Y A034029 Cf. A034027-. %K A034029 nonn %O A034029 0,2 %A A034029 njas %I A034031 %S A034031 1,3,6,11,17,18,33,34,38,43,51,57,66,67,82,83,86,89,102,113,114,118, %T A034031 123,129,131,139,146,171,177,179,187,193,194,198,201,209,214,219, %U A034031 227,233,241,242,246,249,257,258,262,267,274,283,291,307,321,323 %N A034031 Primitively but not imprimitively represented by x^2+2y^2. %Y A034031 Cf. A034027-. %K A034031 nonn %O A034031 0,2 %A A034031 njas %I A065504 %S A065504 1,3,6,11,17,24,33,43,54,66,79,94,110,127,145,164,184,206,229,253,278, %T A065504 304,331,359,389,420,452,485,519,554,590,627,665,705,746,788,831,875, %U A065504 920,966,1013,1061,1110,1161,1213,1266,1320,1375,1431,1488,1546,1605 %N A065504 a(n+1) = a(n) + n + the number of a(k)'s <= n, 1 <= k <= n and a(1) = 1. %e A065504 a(7) = a(6) + 6 + the number of a(k)'s <= 6 = 24 + 6 + 3 ( a(3) is <= 6) = 33. %t A065504 f[1] = 1; f[n_] := f[n] = Block[ {k = 1}, While[ f[k] < n, k++ ]; k--; Return[ f[n - 1] + (n - 1) + k]]; Table[ f[n], {n, 1, 60} ] %K A065504 nonn %O A065504 1,2 %A A065504 Robert G. Wilson v (rgwv@kspaint.com), Nov 25 2001 %I A025735 %S A025735 1,3,6,11,17,24,33,43,54,67,81,97,114,132,152,173,195,219,244,270,298, %T A025735 327,358,390,423,458,494,531,570,610,651,694,738,784,831,879,929,980, %U A025735 1032,1086,1141,1197,1255,1314,1375,1437,1500,1565,1631,1698,1767,1837 %N A025735 Index of 9^n within sequence of numbers of form 5^i*9^j. %K A025735 nonn %O A025735 1,2 %A A025735 dww %I A023601 %S A023601 1,3,6,11,17,24,33,44,56,69,84,101,120,140,161,184,209,236,265,295, %T A023601 326,359,394,431,470,511,553,596,641,688,737,788,841,896,952,1009, %U A023601 1068,1129,1192,1257,1324,1393,1464,1536,1609,1684,1761,1840,1921 %N A023601 Convolution of A023532 and odd numbers. %K A023601 nonn %O A023601 1,2 %A A023601 Clark Kimberling (ck6@cedar.evansville.edu) %I A003022 M2540 %S A003022 1,3,6,11,17,25,34,44,55,72,85,106,127,151,177,199,216,246,283,333,356, %T A003022 372 %N A003022 Shortest Golomb ruler with n marks. %C A003022 a(n) is the least integer such that there is an n-element set of integers between 0 and a(n), the sums of pairs (of not necessarily distinct elements) of which are distinct. %D A003022 CRC Handbook of Combinatorial Designs, 1996, p. 315. %D A003022 A. K. Dewdney, Computer Recreations, Scientific Amer. 253 (No. 6, June), 1985, pp. 16ff; 254 (No. 3, March), 1986, pp. 20ff. %D A003022 S. W. Golomb, How to number a graph, pp. 23-37 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972. %D A003022 A. Kotzig and P. J. Laufer, Sum triangles of natural numbers having minimum top, Ars. Combin. 21 (1986), 5-13. %H A003022 Anonymous, In Search Of The Optimal 20, 21 and 22 Mark Golomb Rulers %H A003022 Distributed.Net, Project OGR %H A003022 L. Miller, Golomb Rulers %H A003022 B. Rankin, Golomb Ruler Calculations %H A003022 J. B. Shearer, Golomb ruler table %H A003022 N. J. A. Sloane, First few optimal Golomb rulers %H A003022 D. Vanderschel et al., In Search Of The Optimal 20, 21 and 22 Mark Golomb Rulers %H A003022 E. W. Weisstein, Link to a section of The World of Mathematics. %e A003022 a(4)=11 because 0-1-4-9-11 (0-2-7-10-11) resp. 0-3-4-9-11 (0-2-7-8-11) are shortest: there is no b0-b1-b2-b3-b4 with different distances |bi-bj| and max. |bi-bj| < 11 %Y A003022 Cf. A036501, A039953. %Y A003022 0-1-4-9-11 corresponds to 1-3-5-2 in A039953: 0+1+3+5+2=11 %K A003022 nonn,hard,nice %O A003022 2,2 %A A003022 njas %I A025722 %S A025722 1,3,6,11,17,25,34,44,56,69,84,100,117,136,156,178,201,225,251,278,307, %T A025722 337,368,401,435,471,508,546,586,627,670,714,759,806,854,904,955,1007, %U A025722 1061,1116,1173,1231,1290,1351,1413,1477,1542,1608,1676,1745,1816,1888 %N A025722 Index of 7^n within sequence of numbers of form 4^i*7^j. %K A025722 nonn %O A025722 1,2 %A A025722 dww %I A022775 %S A022775 1,3,6,11,17,25,34,44,56,69,84,100,117,136,156,178,201,226,252,279, %T A022775 308,338,370,403,437,473,510,549,589,631,674,718,764,811,860,910, %U A022775 961,1014,1068,1124,1181,1239,1299,1360,1423,1487,1553,1620,1688 %N A022775 Place where n-th 1 occurs in A007336. %K A022775 nonn %O A022775 1,2 %A A022775 Clark Kimberling (ck6@cedar.evansville.edu) %I A025743 %S A025743 1,3,6,11,17,25,34,45,57,70,85,101,119,138,159,181,204,229,255,283,312, %T A025743 343,375,408,443,479,517,556,597,639,682,727,773,821,870,921,973,1026, %U A025743 1081,1137,1195,1254,1315,1377,1440,1505,1571,1639,1708,1779,1851,1924 %N A025743 Index of 10^n within sequence of numbers of form 5^i*10^j. %K A025743 nonn %O A025743 1,2 %A A025743 dww %I A022338 %S A022338 1,3,6,11,17,25,34,45,57,71,86,103,121,141,162,184,208,233,260,288,318, %T A022338 349,382,416,452,489,528,568,610,653,697,743,790,839,889,941,994,1049, %U A022338 1105,1163,1222,1283,1345,1408,1473,1539,1607,1676,1747,1819,1893,1968 %N A022338 Index of 5^n within sequence of numbers of form 3^i*5^j. %K A022338 nonn %O A022338 1,2 %A A022338 Clark Kimberling (ck6@cedar.evansville.edu) %I A000603 M2541 N1004 %S A000603 1,3,6,11,17,26,35,45,58,73,90,106,123,146,168,193,216,243,271,302,335, %T A000603 365,402,437,473,516,557,600,642,687,736,782,835,886,941,999,1050,1111, %U A000603 1167,1234,1297,1357,1424,1491,1564,1636,1703,1778,1852,1931,2012,2095 %N A000603 Nonnegative solutions to x^2 + y^2 <= n. %D A000603 H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63. %K A000603 nonn %O A000603 0,2 %A A000603 njas %E A000603 More terms from dww, May 22, 2000 %I A003453 M2542 %S A003453 1,3,6,11,17,26,36,50,65,85,106,133,161,196,232,276,321,375, %T A003453 430,495,561,638,716,806,897,1001,1106,1225,1345,1480,1616, %U A003453 1768,1921,2091,2262,2451,2641,2850,3060,3290,3521,3773,4026 %N A003453 Dissections of a polygon. %D A003453 P. Lisonek, Closed forms for the number of polygon dissections. Journal of Symbolic Computation 20 (1995), 595-601. %D A003453 R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388. %H A003453 N. J. A. Sloane, Transforms %F A003453 G.f.: (1+x-x^2) / ((1-x)^4*(1+x)^2). %Y A003453 John Layman (layman@calvin.math.vt.edu) observes that this appears to be the alternating sum transform (PSUMSIGN) of A005744. %Y A003453 Cf. A005744. %K A003453 nonn,nice %O A003453 5,2 %A A003453 njas, sp. %I A011901 %S A011901 0,0,0,0,1,3,6,11,17,26,37,52,69,90,114,143,176,214,257, %T A011901 306,360,420,486,559,639,726,821,923,1034,1153,1282,1419, %U A011901 1566,1722,1889,2066,2254,2453,2664,2886,3120,3366,3625 %N A011901 [ n(n-1)(n-2)/19 ]. %K A011901 nonn %O A011901 0,6 %A A011901 njas %I A013932 %S A013932 1,3,6,11,17,34,58,87,123,166,215,274,305,407,482,521,562,647,791, %T A013932 899,1073,1261,1327,1394,1463,1533,1677,1751,1906,1985,2067,2235, %U A013932 2321,2497,2681,2870,2967,3170,3273,3378 %N A013932 a(n) is square-free and is sum of first k square-frees for some k. %Y A013932 Cf. A005117, A013930, A013931. %K A013932 nonn %O A013932 1,2 %A A013932 Henri LIFCHITZ (100637.64@CompuServe.COM) %I A010335 %S A010335 3,6,11,18,22,27,31,38,46,51,59,66,83,102,114,118,123,127,131,139, %T A010335 146,158,162,166,171,179,187,191,198,206,214,227,239,243,251,258,262, %U A010335 278,291,307,311,326,334,354,358,363,367,379,383,387 %N A010335 Central term in continued fraction for sqrt(n) is [ sqrt(n) ]. %K A010335 nonn %O A010335 1,1 %A A010335 njas, Walter Gilbert %I A024667 %S A024667 1,3,6,11,18,25,33,44,57,68,81,99,116,134,152,177,200,223,246,276,304, %T A024667 331,360,397,433,465,501,541,579,617,662,707,749,793,845,895,944,995, %U A024667 1051,1105,1161,1214,1279,1337,1397,1456,1528,1591,1657,1722,1799,1870 %N A024667 a(n) = position of 2*n^3 in A003325. %K A024667 nonn %O A024667 1,2 %A A024667 Clark Kimberling (ck6@cedar.evansville.edu) %I A025210 %S A025210 0,1,3,6,11,18,25,36,50,64,81,104,127,153,182,215,251,292,336,385,438,487, %T A025210 549,616,687,753,835,910,1002,1087,1190,1285,1400,1505,1632,1748,1870,1997 %N A025210 s(n) = [ S3/S1 ]; S3 and S1 = 3rd and 1st elementary symmetric functions of {log(k)}, k = 1,2..n. %K A025210 nonn %O A025210 4,3 %A A025210 Clark Kimberling (ck6@cedar.evansville.edu) %I A010000 %S A010000 1,3,6,11,18,27,38,51,66,83,102,123,146,171,198,227,258, %T A010000 291,326,363,402,443,486,531,578,627,678,731,786,843,902, %U A010000 963,1026,1091,1158,1227,1298,1371,1446,1523,1602,1683 %N A010000 a(0)=1, a(n)=n^2 + 2, n >= 1. %K A010000 nonn %O A010000 0,2 %A A010000 njas %I A014125 N1005 %S A014125 1,3,6,11,18,27,39,54,72,94,120,150,185,225,270,321,378,441,511,588, %T A014125 672,764,864,972,1089,1215,1350,1495,1650,1815,1991,2178,2376,2586, %U A014125 2808,3042,3289,3549,3822,4109,4410,4725,5055,5400,5760,6136,6528,6936 %N A014125 Bisection of A001400. %D A014125 H. R. Henze and C. M. Blair, The number of structurally isomeric hydrocarbons of the ethylene series, J. Amer. Chem. Soc., 55 (1933), 680-685. %D A014125 L. Smiley, Hidden Hexagons, (preprint) %F A014125 G.f.: 1/((1-x)^3*(1-x^3)) %Y A014125 Cf. A014126, A000631. %K A014125 nonn,easy %O A014125 0,2 %A A014125 njas %E A014125 More terms from James A. Sellers (sellersj@math.psu.edu), Dec 24 1999 %I A011849 %S A011849 0,0,0,0,1,3,6,11,18,28,40,55,73,95,121,151,186,226,272, %T A011849 323,380,443,513,590,674,766,866,975,1092,1218,1353,1498, %U A011849 1653,1818,1994,2181,2380,2590,2812,3046,3293,3553,3826 %N A011849 [ C(n,3)/3 ]. %K A011849 nonn %O A011849 0,6 %A A011849 njas %I A014284 %S A014284 1,3,6,11,18,29,42,59,78,101,130,161,198,239,282,329,382,441,502, %T A014284 569,640,713,792,875,964,1061,1162,1265,1372,1481,1594,1721,1852, %U A014284 1989,2128,2277,2428,2585,2748,2915,3088,3267,3448,3639,3832,4029 %N A014284 Partial sums of primes (starting with 1). %Y A014284 Cf. A007504. %K A014284 nonn %O A014284 1,2 %A A014284 Deepan Majmudar (dmajmuda@esq.com) %I A026905 %S A026905 1,3,6,11,18,29,44,66,96,138,194,271,372,507,683,914,1211,1596, %T A026905 2086,2713,3505,4507,5762,7337,9295,11731,14741,18459,23024,28628, %U A026905 35470,43819,53962,66272,81155,99132,120769,146784,177969,215307 %N A026905 a(n) = number of sums S of positive integers satisfying S <= n. %H A026905 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 800 %F A026905 Jeff Burch (gburch@erols.com) points out that this is just the partial sums of the partition numbers. %t A026905 Table[ Sum[ PartitionsP[k], {k, 1, n}], {n, 1, 45}] %K A026905 nonn %O A026905 1,2 %A A026905 Clark Kimberling, ck6@cedar.evansville.edu %I A053992 %S A053992 1,1,3,6,11,18,31,49,78,119,180,267,394,567,813,1151,1616,2244,3099, %T A053992 4240,5769,7790,10462,13965,18552,24502,32223,42176,54972,71340,92242, %U A053992 118800,152481,195017,248621,315945,400315,505694,637068,800380 %N A053992 The number phi_3(n) of Frobenius partitions that allow up to 3 repetitions of an integer in a row. %D A053992 Andrews, George E., Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984. %F A053992 Generating function in Andrews' Memoir %K A053992 easy,nonn %O A053992 1,3 %A A053992 James A. Sellers (sellersj@math.psu.edu), Apr 04 2000 %I A052825 %S A052825 0,0,1,3,6,11,18,31,50,85,144,251,438,789,1420,2601,4792,8907,16618, %T A052825 31219,58814 %N A052825 A simple grammar. %H A052825 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 790 %F A052825 G.f.: -x/(-1+x)*Sum(numtheory[phi](j[1])/j[1]*ln((x^j[1]-1)/(2*x^j[1]-1)),j[1]=1 .. infinity) %p A052825 spec:= [S,{B=Cycle(C),C=Sequence(Z,1 <= card),S=Prod(C,B)},unlabelled]: seq(combstruct[count](spec,size=n),n=0..20); %K A052825 easy,nonn %O A052825 0,4 %A A052825 encyclopedia@pommard.inria.fr, Jan 25 2000 %I A003082 M2543 %S A003082 1,1,3,6,11,18,32,48,75,111,160,224,313,420,562,738,956,1221,1550,1936, %T A003082 2405,2958,3609,4368,5260,6279,7462,8814,10356,12104,14093,16320,18834, %U A003082 21645,24783,28272,32158,36442,41187,46410,52151,58443,65345,72864 %N A003082 Multigraphs with 4 nodes and n edges. %D A003082 CRC Handbook of Combinatorial Designs, 1996, p. 650. %D A003082 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.19). %F A003082 G.f. : (x^8-x^7+x^6+x^4+x^2-x+1)/((x-1)^6*(x+1)^2*(x^2+1)*(x^2+x+1)^2). %Y A003082 Cf. A001399, A014395-A014398. %K A003082 easy,nonn,nice %O A003082 0,3 %A A003082 njas %E A003082 Entry improved by comments from Vladeta Jovovic (vladeta@Eunet.yu), Dec 23 1999 %I A058053 %S A058053 1,1,3,6,11,18,33,49,78,117,171,242,346,469,640,855,1127,1463,1896, %T A058053 2405,3045 %N A058053 3-rowed binary matrices with n ones, up to row and column permutation. %H A058053 Expansion of generating function. %F A058053 a(n)= coefficient of x^n*y^n in expansion of 1 / 3!*(1 / 1 - x) / (1 - x*y)^3 / (1 - x*y^2)^3 / (1 - x*y^3) + 3 / (1 - x) / (1 - x*y) / (1 - x*y^2) / (1 - x*y^3) / (1 - x^2*y^2) / (1 - x^2*y^4) + 2 / (1 - x) / (1 - x*y^3) / (1 - x^3*y^3) / (1 - x^3*y^6)). %Y A058053 Cf. A002727, A049311. %K A058053 more,nonn %O A058053 0,3 %A A058053 Vladeta Jovovic (vladeta@Eunet.yu), Nov 19 2000 %I A004133 M2544 %S A004133 1,3,6,11,19,31,43,63,80 %N A004133 Additive bases: a(n) is the least integer such that there is an n-element set of non-negative integers, the sums of pairs (of distinct elements) of which are distinct and at most a(n). %D A004133 R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404 (v_alpha). %Y A004133 Cf. A004135, A004136. %K A004133 nonn,nice %O A004133 2,2 %A A004133 njas %I A050228 %S A050228 1,3,6,11,19,31,49,76,116,175,262,390,578,854,1259,1853,2724,4001,5873, %T A050228 8617,12639,18534,27174,39837,58396,85596,125460,183884,269509,394999, %U A050228 578914,848455,1243487,1822435,2670925,3914448,5736920,8407883 %N A050228 a(n) is the number of subsequences {s(k)} of {1,2,3,...n} such that s(k+1)-s(k) is 1 or 3. %C A050228 The second differences c(n) of {a(n)} satisfy c(n)=c(n-1)+c(n-3) and give A000930 with the first 5 terms deleted. %K A050228 nonn %O A050228 1,2 %A A050228 John W. Layman (layman@math.vt.edu), Dec 20 1999 %I A001976 M2545 N1006 %S A001976 1,3,6,11,19,32,48,71,101,141,188,249,322,414,518,645,791,966 %N A001976 Restricted partitions. %D A001976 A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 276-281. %K A001976 nonn %O A001976 0,2 %A A001976 njas %I A001911 M2546 N1007 %S A001911 0,1,3,6,11,19,32,53,87,142,231,375,608,985,1595,2582,4179,6763,10944, %T A001911 17709,28655,46366,75023,121391,196416,317809,514227,832038,1346267, %U A001911 2178307,3524576,5702885,9227463,14930350,24157815,39088167,63245984 %N A001911 Fibonacci numbers - 2. %D A001911 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 233. %D A001911 D. G. Rogers, An application of renewal sequences to the dimer problem, pp. 142-153 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979. %H A001911 D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory %F A001911 G.f.: (x+x^2)/(1-2*x+x^3). %F A001911 a(n) = a(n-1) + a(n-2) + 2, a(0)=0, a(1)=1. %Y A001911 A001911(n) = A000045(n+3)-2. %Y A001911 Partial sums of F(n+1)=A000045(n+1) where F(x)=(x+1)st Fibonacci number. %K A001911 nonn,easy,nice %O A001911 0,3 %A A001911 njas %E A001911 More terms and better description from Michael Somos (somos@grail.cba.csuohio.edu) %I A020957 %S A020957 3,6,11,19,32,54,89,147,240,392,637,1035,1678,2720,4405,7133,11546, %T A020957 18688,30243,48941,79194,128146,207351,335509,542872,878394,1421279, %U A020957 2299687 %N A020957 Sum of [ 2*tau^(n-k) ] for k from 1 to infinity. %D A020957 C. Kimberling, Problem 10520 in Amer. Math. Mon. 103 (1996) p. 347. %K A020957 nonn %O A020957 1,1 %A A020957 Clark Kimberling (ck6@cedar.evansville.edu) %I A055417 %S A055417 1,3,6,11,20,36,63,106,171,265,396,573,806,1106,1485,1956,2533,3231, %T A055417 4066,5055,6216,7568,9131,10926,12975,15301,17928,20881,24186,27870, %U A055417 31961,36488,41481,46971,52990,59571,66748,74556,83031,92210,102131 %N A055417 Points in N^n of norm <= 2. %K A055417 nonn %O A055417 0,2 %A A055417 dww %I A018918 %S A018918 3,6,11,20,36,64,113,199,350,615,1080,1896,3328,5841,10251,17990, %T A018918 31571,55404,97228,170624,299425,525455,922110,1618191,2839728, %U A018918 4983376,8745216,15346785,26931731,47261894,82938843 %N A018918 Define the sequence L(a_0,a_1) by a_{n+2} is the greatest integer such that a_{n+2}/a_{n+1}= 0. This is L(3,6). %D A018918 D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences,Advances in Number Theory ( Kingston ON,1991) 333-340,Oxford Sci. Publ.,Oxford Univ. Press,New York,1993;. %K A018918 nonn %O A018918 0,1 %A A018918 rkg@cpsc.ucalgary.ca (Richard Guy) %I A054887 %S A054887 1,3,6,11,20,36,64,113,200,354,626,1107,1958,3464,6128,10839,19172, %T A054887 33913,59988,106111,187696,332009,587280,1038820,1837534,3250353, %U A054887 5749442,10169998,17989372,31820803,56286764,99563792,176115092 %N A054887 Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3,Pi/5,Pi/7). %C A054887 The layer sequence is the sequence of the cardinalities of the layers accumulating around a ( finite-sided ) polygon of the tessellation under successive side-reflections; see the illustration accompanying A054888. %F A054887 G.f.: (1+x)*(x^3-1)*(x^5-1)*(x^7-1)/(x^16-2*x^15+x^12+x^10-x^4-x^6+2*x-1 %K A054887 nonn %O A054887 1,2 %A A054887 Paolo Dominici (pd@full-service.it), May 23 2000 %I A019302 %S A019302 0,1,3,6,11,20,36,64,115,216,430,892,1872,3888,7920,15840, %T A019302 31315,61744,122418,245348,497650,1019032,2096680,4312224, %U A019302 8826320,17925376,36070128,71915616,142239056,279671360 %N A019302 Binomial transform of Thue-Morse sequence A010060. %H A019302 N. J. A. Sloane, Transforms %K A019302 nonn,nice %O A019302 0,3 %A A019302 Jonas Wallgren (jwc@ida.liu.se) %I A018075 %S A018075 1,1,3,6,11,20,36,66,121,220,401,730,1331,2423,4414,8039, %T A018075 14641,26663,48558,88433,161051,293299,534145,972765,1771561, %U A018075 3226296,5875603,10700415,19487171,35489261,64631634,117704565 %N A018075 Powers of fourth root of 11 rounded down. %K A018075 nonn %O A018075 0,3 %A A018075 njas %I A052467 %S A052467 0,1,3,6,11,20,37,70,134,255,476,869 %N A052467 Inverse binomial transform of b(n)=1 for prime n and b(n)=0 for composite n. %H A052467 E. W. Weisstein, Link to a section of The World of Mathematics. %K A052467 nonn,more %O A052467 1,3 %A A052467 Eric W. Weisstein (eric@weisstein.com) %I A006127 M2547 %S A006127 1,3,6,11,20,37,70,135,264,521,1034,2059,4108,8205,16398,32783,65552, %T A006127 131089,262162,524307,1048596,2097173,4194326,8388631,16777240,33554457,67108890 %N A006127 2^n + n. %D A006127 John H. Conway, Richard K. Guy, The Book of Numbers, Copernicus Press, p. 84. %H A006127 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 435 %K A006127 nonn %O A006127 0,2 %A A006127 njas %I A007707 %S A007707 3,6,11,21,35,51,73,98,130,167,204,249,296,347,406,471,538,608,686, %T A007707 768,855,950 %N A007707 prime(n) * ... * prime(a(n)) is product of minimal string of consecutive primes which is abundant. %Y A007707 Cf. A007708, A007741. %K A007707 nonn %O A007707 1,1 %A A007707 wnissen@tfn.net (Walter Nissen) %I A018174 %S A018174 1,1,3,6,11,21,38,70,130,239,441,810,1490,2740,5037,9261, %T A018174 17025,31300,57542,105787,194481,357537,657301,1208394, %U A018174 2221532,4084101,7508277,13803340,25376285,46652176,85766121 %N A018174 Powers of fifth root of 21 rounded down. %K A018174 nonn %O A018174 0,3 %A A018174 njas %I A050951 %S A050951 3,6,11,21,38,77,83,93,227,237,437,453,1133,1253 %N A050951 Discriminants of real quadratic fields with class number 1 and related continued fraction period length of 2. %D A050951 R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A. %Y A050951 Cf. A050950-A050969, A051962-A051965. %K A050951 nonn,fini,full %O A050951 0,1 %A A050951 njas, Jan 04 2000 %I A024495 %S A024495 1,3,6,11,21,42,85,171,342,683,1365,2730,5461,10923,21846,43691, %T A024495 87381,174762,349525,699051,1398102,2796203,5592405,11184810,22369621, %U A024495 44739243,89478486,178956971,357913941,715827882,1431655765,2863311531 %N A024495 C(n,2) + C(n,5) + ... + C(n,3[n/3]+2). %D A024495 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, @nd. ed., Problem 38, p. 70. %F A024495 G.f.: 1/((1-x)^3-x^3). %F A024495 a(n) = (1/3)*(2^n+2*cos( (n-4)*Pi/3 )). %F A024495 a(n) =2*a(n-1)+A010892(n-2) =a(n-1)+A024494(n-1). With initial zero, binomial transform of A011655 which is effectively A010892 unsigned. - Henry Bottomley (se16@btinternet.com), Jun 04 2001 %K A024495 nonn,easy %O A024495 2,2 %A A024495 Clark Kimberling (ck6@cedar.evansville.edu) %I A018177 %S A018177 1,1,3,6,11,22,40,75,140,260,484,898,1666,3092,5738,10648, %T A018177 19758,36663,68033,126242,234256,434685,806602,1496732, %U A018177 2777338,5153632,9563083,17745264,32928125,61101454,113379904 %N A018177 Powers of fifth root of 22 rounded down. %K A018177 nonn,easy %O A018177 0,3 %A A018177 njas %I A024506 %S A024506 0,0,0,0,0,0,1,3,6,11,22,43,82,158,306,591,1146,2225,4327,8426,16432,32082, %T A024506 62707,122687,240263,470916,923720,1813228,3561699,7000581,13767809 %N A024506 a(n) = [ C(2n,n)/2^(n+3) ]. %K A024506 nonn %O A024506 0,8 %A A024506 Clark Kimberling (ck6@cedar.evansville.edu) %I A045693 %S A045693 0,0,1,0,1,3,6,11,23,44,91,179,364,723,1457,2902,5827,11633,23310 %N A045693 Binary words of length n (beginning 0) with autocorrelation function 2^(n-1)+3. %F A045693 a[ 2n-1 ] = 2 a[ 2n-2 ] + 2 a[ n ] for n >= 3; a[ 2n ] = 2 a[ 2n-1 ] +a[ n ] - a[ n+1 ] for n >= 3 %K A045693 nonn %O A045693 1,6 %A A045693 TORSTEN.SILLKE@LHSYSTEMS.COM %I A051284 %S A051284 3,6,11,23,44,92,178,370,719,1487,2897,5969,11651,22223,45083,89516, %T A051284 181385,353683,722589,1423078,2903564,5696576,11635316,22866150, %U A051284 46704206 %N A051284 Relative minima of n/c(n), where c(n) is Hofstadter-Conway sequence A004001. %C A051284 The ratio of n/c(n) (where c(n)=A004001) reaches a maximum of 2.0 when n is a power of 2. When n=6 the ratio has a relative minimum of 1.5, so a(2) = 6. %Y A051284 Cf. A004001. %K A051284 nonn %O A051284 1,1 %A A051284 Jud McCranie (jud.mccranie@mindspring.com) %I A001867 M2548 N1008 %S A001867 1,3,6,11,24,51,130,315,834,2195,5934,16107,44368,122643,341802,956635, %T A001867 2690844,7596483,21524542,61171659,174342216,498112275,1426419858, %U A001867 4093181691,11767920118,33891544419,97764131646,282429537947,817028472960 %N A001867 n-bead necklaces with 3 colors. %D A001867 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. %D A001867 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 162. %D A001867 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a). %H A001867 Index entries for sequences related to necklaces %H A001867 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 3 %F A001867 (1/n)*Sum_{d|n} phi(d)*3^(n/d), n>0. %p A001867 A001867:=proc(n) local d,s; if n = 0 then RETURN(1); else s:=0; for d in divisors(n) do s:=s+phi(d)*3^(n/d); od; RETURN(s/n); fi; end; %p A001867 spec:=[N, {N=Cycle(bead), bead=Union(R,G,B), R=Atom, B=Atom, G=Atom}]; [seq(combstruct[count](spec, size=n), n=1..40)]; %K A001867 nonn,easy,nice %O A001867 0,2 %A A001867 njas %I A000998 M2549 N1009 %S A000998 1,3,6,11,24,69,227,753,2451,8004,27138,97806,375313,1511868, %T A000998 6292884,26826701,116994453,523646202,2414394601,11487130362,56341183365 %N A000998 From a differential equation. %D A000998 S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767. %K A000998 nonn %O A000998 0,2 %A A000998 njas %I A038587 %S A038587 3,6,12,12,18,21,27,27,30,36,42,42,48,48,54,54,63,69,69, %T A038587 69,75,78,84,84,90,96,102,102,102,102,114,114,120,123,129, %U A038587 129,135,141,141,141,144,150,156,156,168,168,174,174,174 %N A038587 Sizes of successive clusters in hexagonal lattice A_2 centered at deep hole. %C A038587 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. %H A038587 G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2 %F A038587 Partial sums of A005882. %Y A038587 Cf. A038588. %K A038587 nonn %O A038587 0,1 %A A038587 njas %I A062053 %S A062053 3,6,12,13,24,26,48,52,53,96,104,106,113,192,208,212,213,226,227,384,416, %T A062053 424,426,452,453,454,768,832,848,852,853,904,906,908,909,1536,1664,1696, %U A062053 1704,1706,1808,1812,1813,1816,1818,3072,3328,3392,3408,3412,3413,3616 %N A062053 Numbers with 3 odd integers in their Collatz (or 3x+1) trajectory. %C A062053 The Collatz (or 3x+1) function is f(x) = x/2 if x is even, 3x+1 if x is odd. %C A062053 The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached. %C A062053 Sequence is 2-automatic. %D A062053 J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (Feb 1992) pp. 182-185. %H A062053 Index entries for sequences related to 3x+1 (or Collatz) problem %Y A062053 Cf. A062052-A062060. %K A062053 nonn %O A062053 1,1 %A A062053 dww %E A062053 The Collatz trajectory of 3 is (3,10,5,16,8,4,2,1), which contains 3 odd integers. %I A016051 %S A016051 3,6,12,15,21,24,30,33,39,42,48,51,57,60,66,69,75,78,84,87,93,96, %T A016051 102,105,111,114,120,123,129,132,138,141,147,150,156,159,165,168, %U A016051 174,177,183,186,192,195,201,204,210,213,219,222,228,231,237,240 %N A016051 Of the form 9n+3 or 9n+6. %C A016051 a(n+1) = a(n) + its digital root. %Y A016051 Equals 3*A001651. %K A016051 nonn,base,easy %O A016051 2,1 %A A016051 Robert G. Wilson v (rgwv@kspaint.com) %I A016052 %S A016052 3,6,12,15,21,24,30,33,39,51,57,69,84,96,111,114,120,123,129,141, %T A016052 147,159,174,186,201,204,210,213,219,231,237,249,264,276,291,303, %U A016052 309,321,327,339,354,366,381,393,408,420,426,438,453,465,480,492 %N A016052 a(n+1) = a(n) + sum of its digits. %D A016052 A Result and a Conjecture on Digit Sum Sequences, G. E. Stevens and L. G. Hunsberger, J. Recreational Math. 27, no. 4 (1995), pp. 285-288. %Y A016052 Cf. A004207. %K A016052 nonn,base,easy %O A016052 2,1 %A A016052 Robert G. Wilson v (rgwv@kspaint.com) %I A032602 %S A032602 3,6,12,16,24,28,38,44,54,62,68,80,90,94,110,120,128,134,142,150, %T A032602 160,172,182,194,208,216,230,236,242,248,268,282,296,302,318,322, %U A032602 346,356,362,374,384,392,410,416,428,434,448,464,486,490,500,512 %N A032602 a(n) = n-th prime number + n-th lucky number. %Y A032602 Cf. A000959. %K A032602 nonn %O A032602 0,1 %A A032602 Patrick De Geest (pdg@worldofnumbers.com), may 1998. %I A038046 %S A038046 1,1,3,6,12,17,32,39,63,81,120,131,213,226,311,377,503,520,742,761, %T A038046 1031,1169,1442,1465,2008,2093,2558,2801,3465,3494,4591,4622,5628,6054, %U A038046 7111,7390,9321,9358,10899,11616,13873,13914,17070,17113,20063 %N A038046 Shifts left under transform T where Ta is (identity) DCONV a. %K A038046 nonn,eigen %O A038046 1,3 %A A038046 Christian G. Bower (bowerc@usa.net) %I A038588 %S A038588 3,6,12,18,21,27,30,36,42,48,54,63,69,75,78,84,90,96,102, %T A038588 114,120,123,129,135,141,144,150,156,168,174,180,186,192, %U A038588 198,207,213,219,225,231,240,246,252,258,270,276,282 %N A038588 Sizes of clusters in hexagonal lattice A_2 centered at deep hole. %C A038588 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. %H A038588 G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2 %F A038588 unique(A038587). %Y A038588 Cf. A005882, A038587. %K A038588 nonn %O A038588 0,1 %A A038588 njas %I A028882 %S A028882 3,6,12,18,24,36,48,54,60,66,90,96,108,132,138,156,162,174,186,192, %T A028882 204,216,222,228,240,246,270,306,318,330,360,366,384,390,408,438, %U A028882 450,468,474,480,492,498,522,570,576,582,606,612,618,624,636,642 %N A028882 n^2 - 7 is prime. %H A028882 P. De Geest, Palindromic Quasipronics of the form n(n+x) %K A028882 nonn %O A028882 0,1 %A A028882 Patrick De Geest (pdg@worldofnumbers.com) %I A024513 %S A024513 1,3,6,12,18,25,34,42,55,66,80,94,110,127,144,162,183 %N A024513 a(n) = position of n^2 + (n+1)^2 in A004431 (sums of 2 distinct nonzero squares). %K A024513 nonn %O A024513 1,2 %A A024513 Clark Kimberling (ck6@cedar.evansville.edu) %I A006156 M2550 %S A006156 1,3,6,12,18,30,42,60,78,108,144,204,264,342,456,618,798, %T A006156 1044,1392,1830,2388,3180,4146,5418,7032,9198,11892,15486,20220, %U A006156 26424,34422,44862,58446,76122,99276,129516,168546,219516,285750 %N A006156 Ternary square-free words of length n. %D A006156 F.-J. Brandenburg, Uniformly growing k-th power-free homomorphisms, Theoretical Computer Sci., 23 (1983), 69-82. %D A006156 J. Brinkhuis, Non-repetitive sequences on three symbols, Quart. J. Math. Oxford, 34 (1983), 145-149. %D A006156 John Noonan and Doron Zeilberger, The Goulden-Jackson Cluster Method: Extensions, Applications and Implementations, 1997. %H A006156 M. Baake, V. Elser and U. Grimm, Entropy of Square-Free Words %H A006156 D. Ekhad and D. Zeilberger, There are more than 2^(n/17) n-letter ternary square-free words, J. Integer Sequences, vol. 1, 98.1.9 %H A006156 E. W. Weisstein, Link to a section of The World of Mathematics. %Y A006156 Cf. A060688. %K A006156 nonn,nice %O A006156 0,2 %A A006156 njas,jos, zeilberg@euclid.math.temple.edu (Doron Zeilberger) %I A061776 %S A061776 1,3,6,12,18,30,42,66,90,138,186,282,378,570,762,1146,1530,2298, %T A061776 3066,4602,6138,9210,12282,18426,24570,36858,49146,73722,98298, %U A061776 147450,196602,294906,393210,589818,786426,1179642,1572858,2359290 %N A061776 Start with a single triangle; at n-th generation add a triangle at each vertex, allowing triangles to overlap; sequence gives number of triangles in n-th generation. %D A061776 R. Reed, The Lemming Simulation Problem, Math. in School, 3 (#6, Nov. 1974), 5-6. %D A061776 R. Reed, (illustration to be added). %F A061776 Explicit formula given in Maple line. %p A061776 A061776 :=proc(n) if n mod 2 = 0 then 6*(2^(n/2)-1); else 3*(2^((n-1)/2)-1)+3*(2^((n+1)/2)-1); fi; end; # for n >= 1 %Y A061776 A061777 gives total population of triangles at n-th generation. %K A061776 nonn,nice,easy %O A061776 0,2 %A A061776 njas, rkg, Jun 23 2001 %I A061061 %S A061061 0,0,1,3,6,12,20,31,46,64,87,115,147,186,231,282,342,408,482,566,657, %T A061061 759,871,991,1126,1270,1424,1594,1774,1968,2177,2397,2635,2887,3151, %U A061061 3436,3735,4050,4386,4736,5106,5496,5901,6330,6778,7244,7737,8247,8778 %N A061061 Maximal number of 132 patterns in a permutation of 1,2,...,n. %D A061061 W. Stromquist, Packing layered posets into posets, manuscript. %H A061061 Miklos Bona, Bruce E. Sagan and Vincent R. Vatter, Pattern frequency sequences and internal zeros %F A061061 a(n) = max(a(k) + k*C(n-k,2): 1 <= k < n) %e A061061 a(8) = 31 because the permutation of 1..8 containing the maximum number of 132 patterns is 13287654. %K A061061 easy,nonn %O A061061 1,4 %A A061061 Michael H. Albert (malbert@cs.otago.ac.nz), May 27 2001 %E A061061 More terms from Vladeta Jovovic (vladeta@Eunet.yu), Jun 03 2001 %I A001975 M2551 N1010 %S A001975 1,1,3,6,12,20,32,49,73,102,141,190,252,325,414,521,649,795,967 %N A001975 Restricted partitions. %D A001975 A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 276-281. %K A001975 nonn %O A001975 0,3 %A A001975 njas %I A034333 %S A034333 0,0,1,3,6,12,20,33,52,78,113,163 %N A034333 Matroids: column 3 of A034328. %D A034333 Computed by Harald Fripertinger (fripert@kfunigraz.ac.at). %H A034333 Index entries for sequences related to matroids %H A034333 H. Fripertinger, Isometry Classes of Codes %Y A034333 Cf. A034327-. %K A034333 nonn %O A034333 0,4 %A A034333 njas %I A006128 M2552 %S A006128 0,1,3,6,12,20,35,54,86,128,192,275,399,556,780,1068,1463,1965,2644,3498, %T A006128 4630,6052,7899,10206,13174,16851,21522,27294,34545,43453,54563,68135, %U A006128 84927,105366,130462,160876,198014,242812,297201,362587,441546,536104 %N A006128 Total number of parts in all partitions of n. Also, sum of largest parts of all partitions of n. %C A006128 a(n) = degree of Kac-determinant at level n as polynomial in the conformal weight (called h). (Cf. C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Vol. 2, p.533, eq.(98); ref. p.643, Cambridge University Press, (1989)) - comment from Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de) %F A006128 G.f: Sum n*x^n Product 1/(1-x^k); k = 1..n; n=1..inf. %F A006128 G.f: Sum x^k/(1-x^k); k=1..inf * Product 1/(1-x^m); m=1..inf. %F A006128 a(n) = Sum divisors of m * partitions of n-m; m=1..n. %Y A006128 a(n) = sum(k*A008284(n, k), k=1..n). %K A006128 nonn,easy,nice %O A006128 0,3 %A A006128 njas, clm %I A028926 %S A028926 0,1,3,6,12,20,36,46,66,90,138,216,378,459,616,873,1645, %T A028926 2133 %N A028926 Number of pairs of minimal vectors in n-dimensional lattice Kappa_n. %D A028926 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 161. %Y A028926 A028923 / 2. %K A028926 nonn %O A028926 0,3 %A A028926 njas %I A038577 %S A038577 1,3,6,12,20,36,58,100,160,268,430,708,1140,1860,3002,4876, %T A038577 7880,12772,20654,33444,54100,87564,141666,229252,370920, %U A038577 600196,971118,1571340,2542460,4113828,6656290,10770148 %N A038577 Self-avoiding walks of length n from origin in strip Zx{0,1}. %D A038577 J Labelle, Self-avoiding walks and polyominoes in strips, Bull. ICA, 23 (1998), 88-98. %p A038577 f:=n->if n mod 2 = 0 then 8*fibonacci(n)-n else 8*fibonacci(n)-4; fi; %K A038577 nonn,walk,easy %O A038577 0,2 %A A038577 njas %I A028925 %S A028925 0,1,3,6,12,20,36,63,120,136 %N A028925 Maximal number of pairs of minimal vectors in an n-dimensional lattice. %D A028925 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 15. %Y A028925 A001116 / 2. %K A028925 nonn,hard %O A028925 0,3 %A A028925 njas %I A028924 %S A028924 0,1,3,6,12,20,36,63,120,136,168,219,324,453,711,1170,2160, %T A028924 2673,3699,5334,8700,13860,24948,46575,98280,98328,98424, %U A028924 98571,98868,99253,100023,101346,104160 %N A028924 Maximal number of pairs of minimal vectors in n-dimensional laminated lattice. %D A028924 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 174. %Y A028924 A002336 / 2. %K A028924 nonn,hard %O A028924 0,3 %A A028924 njas, jhc %E A028924 Entry for n=31 corrected by C. Muses. %I A034738 %S A034738 1,3,6,12,20,42,70,144,270,540,1034,2112,4108,8274,16440,32928,65552, %T A034738 131418,262162,524880,1048740,2098206,4194326,8391024,16777300, %U A034738 33558564,67109418,134226120,268435484,536888520,1073741854,2147516736 %N A034738 Dirichlet convolution of b_n=2^(n-1) with phi(n). %F A034738 (1/2)* Sum_{d|n} phi(d)*2^(n/d), n >= 1. %Y A034738 Equals A053635 / 2. Cf. A000740. %K A034738 nonn %O A034738 1,2 %A A034738 Erich Friedman (erich.friedman@stetson.edu) %I A054064 %S A054064 1,1,3,6,12,20,45,63,84,144,180,225,330,396,468,637,735,840,1088,1224, %T A054064 1377,1710,1900,2310,2541,2783,3312,3600,3900,4563,4914,5684,6090,6525, %U A054064 7440,7936,8976,9537,10115,11340,11988 %N A054064 Least k for which the integers Floor(2k/(m*(m+1))) for m=1,2,...,n are distinct. %K A054064 nonn %O A054064 1,3 %A A054064 Clark Kimberling, ck6@cedar.evansville.edu %I A053479 %S A053479 0,0,3,6,12,21,30,42,54,69,90,102,129,150,174,198,225,258,288,327,354, %T A053479 396,435,471,522,558,609,654,702,759,807,864,924,981,1038,1104,1173, %U A053479 1230,1308,1368,1443,1512,1590,1671,1746,1830,1908,2001,2076,2166,2265 %N A053479 Circle numbers (version 6): a(n)= number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2, 1/(2*sqrt(3))). %C A053479 a(n)/(n/2)^2->Pi*2/sqrt(3) %Y A053479 Cf. A053411, A053414, A053415, A053416, A053417. %K A053479 easy,nonn %O A053479 0,3 %A A053479 Klaus Strassburger (strass@dfi.uni-duesseldorf.de), Jan 14 2000 %I A011779 %S A011779 1,3,6,12,21,33,51,75,105,145,195,255,330,420,525,651,798, %T A011779 966,1162,1386,1638,1926,2250,2610,3015,3465,3960,4510, %U A011779 5115,5775,6501,7293,8151,9087,10101,11193,12376,13650 %N A011779 Expansion of 1/((1-x)^3*(1-x^3)^2). %K A011779 nonn %O A011779 0,2 %A A011779 Emeric Deutsch (deutsch@duke.poly.edu) %I A034344 %S A034344 0,0,1,3,6,12,21,34,54,82,120,174,244,337,458,613,808,1056,1361, %T A034344 1738,2200,2759,3431,4240,5198 %N A034344 Binary [ n,3 ] codes without 0 columns. %D A034344 H. Fripertinger and A. Kerber, in AAECC-11, Lect. Notes Comp. Sci. 948 (1995), 194-204. %H A034344 H. Fripertinger, Isometry Classes of Codes %Y A034344 Cf. A034253, A034344-. %K A034344 nonn %O A034344 1,4 %A A034344 njas %I A054578 %S A054578 1,3,6,12,21,35,56,90,139,215,316,462,667,961,1358,1918,2665,3693,5034, %T A054578 6844,9187,12365 %N A054578 Number of subsequences of {1..n} such that all differences of pairs of terms are distinct (i.e. number of Golomb rulers on {1..n}). %Y A054578 Cf. A003022. %K A054578 nonn %O A054578 1,2 %A A054578 John W. Layman (layman@math.vt.edu), Apr 11 2000 %I A006330 M2553 %S A006330 1,1,3,6,12,21,38,63,106 %N A006330 Corners. %D A006330 G. Kreweras, Sur les extensions lineaires d'une famille particuliere d'ordres partiels, Discrete Math., 27 (1979), 279-295. %K A006330 nonn %O A006330 0,3 %A A006330 njas %I A000991 M2554 N1011 %S A000991 1,3,6,12,21,40,67,117,193,319,510,818,1274,1983,3032,4610,6915, %T A000991 10324,15235,22371,32554,47119,67689,96763,137404,194211,272939,381872 %N A000991 3-line partitions of n. %D A000991 L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.8). %D A000991 M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273. %D A000991 P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. %F A000991 G.f.: prod from k =1 to inf ( 1 - x^k )^(-3) / ( 1 - x )^2 ( 1 - x^2 ). %Y A000991 Cf. A000990. %K A000991 nonn %O A000991 1,2 %A A000991 njas %I A062483 %S A062483 1,1,3,6,12,22,35,54,78,109,147,194,249,314,389,476,574,685,809,947, %T A062483 1100,1268,1451,1651,1869,2104,2357,2630,2922,3234,3567,3921,4297,4696, %U A062483 5118,5563,6032,6526,7045,7590,8162,8760,9385,10038,10719,11429 %N A062483 Nearest integer to (Product(n^((1 + ln(i))/i^2), {i, 1, n})). %t A062483 Round[Product[n^((1 + Log[i])/i^2), {i, 1, n}]] %Y A062483 Cf. A062482. %K A062483 nonn %O A062483 0,3 %A A062483 Olivier Gerard (ogerard@ext.jussieu.fr), Jun 23 2001 %I A018078 %S A018078 1,1,3,6,12,22,41,77,144,268,498,928,1728,3216,5985,11141, %T A018078 20736,38594,71831,133693,248832,463128,861979,1604324, %U A018078 2985984,5557542,10343751,19251891,35831808,66690509,124125023 %N A018078 Powers of fourth root of 12 rounded down. %K A018078 nonn %O A018078 0,3 %A A018078 njas %I A005404 M2555 %S A005404 1,3,6,12,22,42,75,135,238,416 %N A005404 Protruded partitions of n. %D A005404 R. P. Stanley, A Fibonacci lattice, Fib. Quart., 13 (1975), 215-232. %K A005404 nonn %O A005404 1,2 %A A005404 njas %I A055244 %S A055244 1,1,3,6,12,23,43,79,143,256,454,799,1397,2429,4203,7242,12432,21271, %T A055244 36287,61739,104791,177476,299978,506111,852457,1433593,2407443, %U A055244 4037454,6762708,11314391,18909139,31569799,52657247,87751624 %N A055244 Number of certain stackings of n+1 squares on a double staircase. %C A055244 a(n)= G_{n+1} of Turban ref. eq.(3.9). %D A055244 L. Turban, Lattice animals on a staircase and Fibonacci numbers, J.Phys. A 33 (2000) 2587-2595. %F A055244 G.f.: (1-x+x^3)/(1-x-x^2)^2 (from Turban ref. eq.(3.3) with t=1). %F A055244 a(n)=((n+5)*F(n+1)+(2*n-3)*F(n))/5 with F(n)=A000045(n) (Fibonacci numbers) (from Turban ref. eq.(3.9)). %Y A055244 A000045, A055245. %K A055244 nonn,easy %O A055244 0,3 %A A055244 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), May 10 2000 %I A018180 %S A018180 1,1,3,6,12,23,43,80,150,282,529,990,1854,3471,6498,12167, %T A018180 22778,42645,79839,149474,279841,523910,980849,1836318, %U A018180 3437902,6436343,12049936,22559543,42235329,79071767,148035889 %N A018180 Powers of fifth root of 23 rounded down. %K A018180 nonn %O A018180 0,3 %A A018180 njas %I A050243 %S A050243 1,1,3,6,12,23,44,84,158,297,561,1056,1989,3746,7056,13287,25023,47125, %T A050243 88746,167127,314735,592710,1116193,2102019,3958531,7454720,14038755, %U A050243 26437833,49787820,93760597,176570285,332517781,626198652,1179259497 %N A050243 [ (1/2 * (sqrt(2) + 1 + sqrt(2*sqrt(2) - 1)))^n ]. %K A050243 nonn,easy %O A050243 0,3 %A A050243 xpolakis@otenet.gr (Antreas P. Hatzipolakis), Dec 21 1999 %E A050243 More terms from dww, Dec 22, 1999. %I A024505 %S A024505 0,0,0,0,1,1,3,6,12,23,45,86,165,317,612,1183,2292,4451,8654,16853, %T A024505 32865,64165,125414,245375,480526,941832,1847440,3626457,7123398, %U A024505 14001162,27535618,54182991,106672764,210113021,414046247,816262602 %N A024505 a(n) = [ C(2n,n)/2^(n+2) ]. %K A024505 nonn %O A024505 0,7 %A A024505 Clark Kimberling (ck6@cedar.evansville.edu) %E A024505 More terms from James A. Sellers (sellersj@math.psu.edu), Feb 06 2000 %I A005256 M2556 %S A005256 1,3,6,12,23,45,87,171,336,666,1320,2628,5233,10443 %N A005256 Weighted voting procedures. %D A005256 Kreweras, G.; Sur quelques problemes relatifs au vote pondere, [ Some problems of weighted voting ] Math. Sci. Humaines No. 84 (1983), 45-63. %D A005256 T. V. Narayana, Recent progress and unsolved problems in dominance theory, pp. 68-78 of Combinatorial mathematics (Canberra 1977), Lect. Notes Math. Vol 686, 1978. %D A005256 T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101. %K A005256 nonn,easy,nice,more %O A005256 1,2 %A A005256 njas %I A003204 M2557 %S A003204 1,3,6,12,24,33,60,99,156,276,438,597 %N A003204 Cluster series for honeycomb. %D A003204 J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol 2; see especially pp. 225-226. %D A003204 M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315. %K A003204 nonn %O A003204 0,2 %A A003204 njas %I A038620 %S A038620 1,3,6,12,24,35,48,69,86,108,138,161,192,231,260,300,348,383,432, %T A038620 489,530,588,654,701,768,843,896,972,1056,1115,1200,1293,1358,1452, %U A038620 1554,1625,1728,1839 %N A038620 Growth function of an infinite cubic graph (no of nodes at distance n from fixed node). %F A038620 a(0) = 1, a(1) = 3, a(2) = 6; a(n) = 4n^2 / 3, n >= 3 and n == 0 (mod 3); a(n) = (4n^2 + n + 4) / 3, n >= 3 and n == 1 (mod 3); a(n) = (4n^2 - n + 10) / 3, n >= 3 and n == 2 (mod 3). %K A038620 nonn %O A038620 0,2 %A A038620 Jan Kristian Haugland (jankrihau@hotmail.com) %I A039695 %S A039695 3,6,12,24,39,51,57,60,75,87,102,105,132,150,174,186,201,219,246,249, %T A039695 276,282,303 %N A039695 Twin Fibonacci-lucky numbers (middle terms). %C A039695 See A039672 for definition. %Y A039695 A039673. %K A039695 nonn %O A039695 1,1 %A A039695 Felice Russo (felice.russo@katamail.com) %I A018183 %S A018183 1,1,3,6,12,24,45,85,161,305,576,1087,2053,3877,7321,13824, %T A018183 26102,49285,93059,175712,331776,626451,1182849,2233426, %U A018183 4217100,7962624,15034827,28388386,53602241,101210414,191102976 %N A018183 Powers of fifth root of 24 rounded down. %K A018183 nonn %O A018183 0,3 %A A018183 njas %I A048719 %S A048719 0,3,6,12,24,48,51,96,99,102,192,195,198,204,384,387,390, %T A048719 396,408,768,771,774,780,792,816,819,1536,1539,1542,1548, %U A048719 1560,1584,1587,1632,1635,1638,3072,3075,3078,3084,3096 %N A048719 Binary expansion matches ((0)*0011)*(0*). %C A048719 1-bits occur only in pairs, separated from other such pairs by at least two 0-bits. %C A048719 All terms satisfy both rule150(n) = 3*n and rule90(n) = 5*n. %F A048719 a(n) = 3*A048718[ n ] %Y A048719 Intersection of A048716 and A048717. %K A048719 nonn %O A048719 0,2 %A A048719 Antti.Karttunen@iki.fi (karttu@megabaud.fi), 30.3.1999 %I A002910 M2558 N1012 %S A002910 1,3,6,12,24,48,90,168,318,600,1098,2004,3696,6792,12270,22140, %T A002910 40224,72888,130650,234012,421176,756624,1348998,2403840,4299018 %N A002910 Susceptibility series for honeycomb. %D A002910 C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 380. %D A002910 M. F. Sykes, Some counting theorems in the theory of the Ising problem and the excluded volume problem, J. Math. Phys., 2 (1961), 52-62. %D A002910 M. F. Sykes and M. E. Fisher, Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices, Physica, 28 (1962), 919-938. %D A002910 M. F. Sykes, D. G. Gaunt, P. D. Roberts and J. A. Wyles, High temperature series for the susceptibility of the Ising model, I. Two dimensional lattices, J. Phys. A 5 (1972) 624-639. %K A002910 nonn,nice %O A002910 0,2 %A A002910 njas %E A002910 Sykes, Gaunt et al. give 8 more terms. %I A001668 M2559 N1013 %S A001668 1,3,6,12,24,48,90,174,336,648,1218,2328,4416,8388,15780,29892, %T A001668 56268,106200,199350,375504,704304,1323996,2479692,4654464,8710212 %N A001668 Self-avoiding n-step walks on honeycomb lattice. %D A001668 M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58. %D A001668 M. F. Sykes, Some counting theorems in the theory of the Ising problem and the excluded volume problem, J. Math. Phys., 2 (1961), 52-62. %D A001668 M. F. Sykes et al., The asymptotic behaviour of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660. %K A001668 nonn,walk,nice,more %O A001668 0,2 %A A001668 njas %E A001668 Sykes et al. give 34 terms. %I A033893 %S A033893 3,6,12,24,48,96,165,321,444,888,1776,3453,6798,13587,27165,39732,63111, %T A033893 74247,98724,123513,235848,470436,504903,508362,531930,545289,790878, %U A033893 868767,1536555,2892111,4004400,4004844,4049292,4273791,5508570,5564148 %N A033893 Sort then Add!. %C A033893 a(n+1) is obtained by sorting the digits of a(n) into increasing order then adding this number to a(n). %e A033893 96+69=165. %Y A033893 Cf. A033860. %K A033893 nonn,base,easy %O A033893 1,1 %A A033893 njas, dww %I A006851 M2560 %S A006851 1,3,6,12,24,48,96,186,360,696,1344,2562,4872,9288,17664,33384,63120, %T A006851 119280,225072,423630,797400,1499256,2817216,5286480,9918768,18592080 %N A006851 Trails of length n on honeycomb. %D A006851 A. J. Guttmann, Lattice trails II: numerical results, J. Phys. A 22 (1989), 575-588. %K A006851 nonn,walk %O A006851 0,2 %A A006851 njas %I A046944 %S A046944 1,3,6,12,24,48,96,192,384,768,1506,2982,5904,11688,23094,45678, %T A046944 90000,177660,349938,690192,1359288,2678808,5271558,10381926 %N A046944 Self-avoiding walks of length n on hydrogen peroxide lattice. %D A046944 J. A. Leu, Self-avoiding walks on a pair of three dimensional lattices, Phys. Lett., 29A (1969), 641-642. %K A046944 nonn,walk,more %O A046944 0,2 %A A046944 njas %E A046944 Reference gives 31 terms. %I A003945 %S A003945 1,3,6,12,24,48,96,192,384,768,1536,3072,6144,12288,24576, %T A003945 49152,98304,196608,393216,786432,1572864,3145728,6291456, %U A003945 12582912,25165824,50331648,100663296,201326592,402653184 %N A003945 Coordination sequence of infinite tree with valency 3. %H A003945 Index entries for sequences related to trees %H A003945 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 151 %H A003945 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 304 %F A003945 G.f.: (1+x)/(1-2*x) %F A003945 a(n)=2a(n-1), n>1; a(0)=1, a(1)=3. %p A003945 k:=3; if n = 0 then 1 else k*(k-1)^(n-1); fi; %Y A003945 Essentially same as A007283 (3*2^n) and A042950. %K A003945 nonn,easy %O A003945 0,2 %A A003945 njas %I A007283 M2561 %S A007283 3,6,12,24,48,96,192,384,768,1536,3072,6144,12288,24576,49152,98304,196608, %T A007283 393216,786432,1572864,3145728,6291456,12582912,25165824,50331648,100663296, %U A007283 201326592,402653184,805306368,1610612736,3221225472,6442450944,12884901888 %N A007283 3*2^n. %C A007283 Same as Pisot sequences E(3,6), L(3,6), P(3,6), T(3,6). See A008776 for definitions of Pisot sequences. %F A007283 G.f.: 3/(1-2*x) %F A007283 a(n)=2a(n-1), n>0; a(0)=3. %o A007283 (PARI) a(n)=3*2^n %Y A007283 Essentially same as A003945 and A042950. %K A007283 easy,nonn %O A007283 0,1 %A A007283 njas %I A049942 %S A049942 1,1,3,6,12,24,48,98,199,393,786,1574,3151,6308,12628,25280,50610, %T A049942 101123,202246,404494,808991,1617988,3235988,6472000,12944050,25888201, %U A049942 51776596,103553585,207107958,414217493,828438143,1656882606 %N A049942 a(n)=a(1)+a(2)+...+a(n-1)+a(m), where m=n-1-2^p, and 2^pHome page for hexagonal (or triangular) lattice A2 %K A007239 nonn %O A007239 1,1 %A A007239 sp %I A004067 %S A004067 1,1,1,3,6,12,26,42 %N A004067 The coding-theoretic function A(n,6,7). %D A004067 A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380. %D A004067 CRC Handbook of Combinatorial Designs, 1996, p. 411. %H A004067 E. M. Rains and N. J. A. Sloane, A(n,d,w) tables %H A004067 Index entries for sequences related to A(n,d,w) %K A004067 nonn,hard %O A004067 7,4 %A A004067 njas %I A054195 %S A054195 1,3,6,12,26,62,157,409,1079,2863,7617,20299,54202,145134,390048, %T A054195 1052840,2855633,7784909,21333806,58769738,162735221,452890963, %U A054195 1266501060,3558037366,10038873751,28437746721,80854303650,230659891380 %N A054195 Binomial transform of A001371. %K A054195 nonn %O A054195 0,2 %A A054195 njas, Apr 29 2000 %I A054190 %S A054190 1,3,6,12,26,62,158,418,1129,3087,8507,23573,65611,183333,514137, %T A054190 1446723,4083725,11560877,32816588,93385330,266360509,761373941, %U A054190 2180721568,6257750542,17988623409,51795127277,149363509694,431341949970 %N A054190 Binomial transform of A001037. %K A054190 nonn %O A054190 0,2 %A A054190 njas, Apr 29 2000 %I A052103 %S A052103 0,1,3,6,12,27,63,144,324,729,1647,3726,8424,19035,43011,97200,219672, %T A052103 496449,1121931,2535462,5729940,12949227,29264247,66134880,149459580, %U A052103 337766841 %N A052103 The third of the three sequences associated with the polynomial x^3 - 2. %D A052103 Ashok Kumar Gupta and Ashok Kumar Mittal, Integer Sequences associated with Integer Monic Polynomial, preprint, Jan., 2000. %F A052103 a(n) = 3 * a(n-1) - 3 * a(n-2) + 3 * a(n-3), n >2. %Y A052103 Cf. A052101, A052102. %K A052103 eigen,frac,nonn %O A052103 0,3 %A A052103 Ashok K. Gupta and Ashok K. Mittal (akgjkiapt@hotmail.com), Jan 20 2000 %I A018011 %S A018011 1,3,6,12,28,63,144,330,755,1728,3957,9058,20736,47474, %T A018011 108688,248832,569684,1304249,2985984,6836197,15650984, %U A018011 35831808,82034362,187811805,429981696,984412343,2253741659 %N A018011 Powers of cube root of 12 rounded up. %K A018011 nonn %O A018011 0,2 %A A018011 njas %I A025208 %S A025208 1,1,3,6,12,28,64,154,382,979,2583,6998,19421,55106,159595,471041,1414958, %T A025208 4320813,13399076,42157076,134459874,434427716,1420861672,4701420879 %N A025208 a(n) = [ (n-2)nd elementary symmetric function of {log(k)} ], k = 2,3,...,n. %K A025208 nonn %O A025208 2,3 %A A025208 Clark Kimberling (ck6@cedar.evansville.edu) %I A049941 %S A049941 1,1,3,6,12,29,55,108,216,539,1025,2024,4031,8056,16109,32216,64432, %T A049941 161079,306051,604049,1204073,2406139,4811279,9622072,19243821, %U A049941 38487534,76975015,153950004,307899991,615799976,1231599949,2463199896 %N A049941 a(n)=a(1)+a(2)+...+a(n-1)+a(m), where m=2^(p+1)+2-n, and 2^pIndex entries for sequences related to Recaman's sequence %H A064290 N. J. A. Sloane, Fortran program for A005132, A057167, A064227, A064228 %e A064290 A005132 begins 1, 3, 6, 2, 7, 13, 20, 12, ... and these terms have heights 1, 2, 3, 2, 3, 4, 5, 4, ... %Y A064290 Cf. A005132, A064288, A064289, A064292, A064293, A064294. %K A064290 nonn %O A064290 1,2 %A A064290 njas, Sep 25 2001 %I A064621 %S A064621 1,3,6,13,20,62,113,224,367,494,833,9169,131313,155719,180118, %T A064621 280766754,3454917187,31685027464 %N A064621 Values of A005132(n) at which the ratio A005132(n)/n sets a new record. %C A064621 See A064622 for associated values of n. %H A064621 Index entries for sequences related to Recaman's sequence %e A064621 A005132(7)=20, 20/7=2.857, larger than the ratio for any smaller value of n. %Y A064621 Cf. A005132, A064622. %K A064621 more,nonn %O A064621 0,2 %A A064621 Jud McCranie (jud.mccranie@mindspring.com), Sep 26 2001 %I A047172 %S A047172 0,0,1,3,6,13,21,45,70,154,235,525,791,1793,2681,6153,9150,21206,31401, %T A047172 73359,108262,254607,374715,886171,1301235,3091971,4531423,10811671, %U A047172 15818791,37877401,55339849,132924649,193962894,467187454 %N A047172 Nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= (n-1)/2. %K A047172 nonn %O A047172 1,4 %A A047172 Clark Kimberling, ck6@cedar.evansville.edu %I A034734 %S A034734 1,3,6,13,21,48,77,163,298,593,1113,2268,4329,8659,17046,33949,67133, %T A034734 134076,266325,531763,1059702,2116065,4222961,8437812,16852321, %U A034734 33680387,67305930,134544781,268949685,537722088,1075088093,2149697251 %N A034734 Dirichlet convolution of b_n=2^(n-1) with Fibonacci numbers. %K A034734 nonn %O A034734 1,2 %A A034734 Erich Friedman (erich.friedman@stetson.edu) %I A019079 %S A019079 0,1,3,6,13,21,51,135,249,543,1328,2956,7043,17203,41428,107072, %T A019079 267949 %N A019079 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite BOG = Boggsite Na4Ca7 [ Al18Si78O192 ]. 74 H2O. %D A019079 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997. %H A019079 G. Thimm, Cycle sequences of crystal structures %K A019079 nonn %O A019079 3,3 %A A019079 Georg Thimm (mgeorg@ntu.edu.sg) %I A048134 %S A048134 3,6,13,22,40,55,88,118,163,205,280,334,436,517,625,733,901,1018, %T A048134 1225,1381,1591,1786,2083,2287,2617,2887,3238,3544,4006,4306,4831, %U A048134 5239,5749,6205,6817,7267,8005,8572,9280,9880,10780,11374,12361 %N A048134 Number of colors that can mixed with up to n units of yellow, blue, red. %F A048134 a(n) = number of triples (i,j,k) with 1 <= i+j+k <= n and gcd(i,j,k) = 1. %e A048134 a(2)=6: primary and secondary colors (Y^1, B^1, R^1, Y^1*B^1, Y^1*R^1, B^1*R^1). %Y A048134 Two colors gives A005728. %Y A048134 Cf. A048240, A048241. %K A048134 nonn,nice,easy %O A048134 1,1 %A A048134 Jurjen N.E. Bos (J.Bos@Interpay-ISS.demon.nl) %E A048134 More terms from Robin Trew (trew@hcs.harvard.edu). %I A058397 %S A058397 1,3,6,13,22,42,66,112,172,270,397,602,858,1245,1748,2464,3381,4671, %T A058397 6302,8537,11372,15147,19914,26201,34057,44250,56986,73277,93497, %U A058397 119161,150809,190590,239496,300388,374912,467135,579394,717384,884813 %N A058397 Row sums of partition triangle A026820. %F A058397 a(n) = sum(A026820(n,k),k=1..n). %Y A058397 A026820. %K A058397 nonn %O A058397 1,2 %A A058397 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de) Dec 11 2000 %I A022811 %S A022811 1,1,3,6,13,23,44,74,129,210,345,542,858,1310,2004,2996,4467,6540, %T A022811 9552,13744,19711,27943,39452,55172,76865,106200,146173,199806, %U A022811 272075,368247,496642,666201,890602,1184957,1571417,2075058,2731677 %N A022811 Number of terms in n-th derivative of a function composed with itself 3 times. %D A022811 W. C. Yang (yang@math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997. %F A022811 If a(n,m) = number of terms in m-derivative of a function composed with itself n times, p(n,k) = number of partitions of n into k parts, then a(n,m)=sum{i=0..m}p(m,i)a(n-1,i). %Y A022811 Cf. A008778, A022812-A022818, A024207-A024210. First column of A039805. %K A022811 nonn %O A022811 0,3 %A A022811 Winston C. Yang (yang@math.wisc.edu) %I A002799 M2563 N1014 %S A002799 1,3,6,13,23,45,78,141,239,409 %N A002799 4-line partitions of n. %D A002799 M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273. %D A002799 P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. %K A002799 nonn %O A002799 1,2 %A A002799 njas %I A058554 %S A058554 1,3,6,13,24,39,64,102,153,230,342,492,704 %N A058554 McKay-Thompson series of class 20E for Monster. %D A058554 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994). %Y A058554 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc. %K A058554 nonn %O A058554 -1,2 %A A058554 njas, Nov 27, 2000 %I A022568 %S A022568 1,3,6,13,24,42,73,120,192,302,465,702,1046,1536,2226,3195, %T A022568 4536,6378,8896,12306,16896,23045,31224,42048,56310,75000, %U A022568 99384,131072,172071,224910,292774,379608,490338,631104 %N A022568 Expansion of Product (1+q^m)^3; m=1..inf. %K A022568 nonn %O A022568 0,2 %A A022568 njas %I A061567 %S A061567 1,1,3,6,13,24,44,75,124,200,314,480,719,1058,1530,2180,3062,4248,5823, %T A061567 7898,10605,14108,18607,24343,31607,40749,52184,66408,84009,105677, %U A061567 132226,164608,203936,251505,308815,377606,459883,557954,674470,812467 %N A061567 Floor[n^ln(n)]. %K A061567 easy,nonn %O A061567 1,3 %A A061567 Leroy Quet (qqquet@mindspring.com), May 18 2001 %I A018081 %S A018081 1,1,3,6,13,24,46,89,169,320,609,1157,2197,4171,7921,15041, %T A018081 28561,54232,102978,195537,371293,705021,1338715,2541992, %U A018081 4826809,9165284,17403307,33045903,62748517,119148698,226242995 %N A018081 Powers of fourth root of 13 rounded down. %K A018081 nonn %O A018081 0,3 %A A018081 njas %I A001452 M2564 N1015 %S A001452 1,3,6,13,24,47,83,152,263,457,768,1292,2118,3462,5564,8888,14016, %T A001452 21973,34081,52552,80331,122078,184161,276303,411870,610818,900721 %N A001452 5-line partitions of n. %D A001452 M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273. %D A001452 P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. %K A001452 nonn %O A001452 1,2 %A A001452 njas %I A005405 M2565 %S A005405 1,3,6,13,24,47,86,159,285,509 %N A005405 Protruded partitions of n. %D A005405 R. P. Stanley, A Fibonacci lattice, Fib. Quart., 13 (1975), 215-232. %K A005405 nonn %O A005405 1,2 %A A005405 njas %I A000219 M2566 N1016 %S A000219 1,1,3,6,13,24,48,86,160,282,500,859,1479,2485,4167,6879,11297, %T A000219 18334,29601,47330,75278,118794,186475,290783,451194,696033,1068745, %U A000219 1632658,2483234,3759612,5668963,8512309,12733429,18974973,28175955 %N A000219 Planar partitions of n. %C A000219 Can also be regarded as number of "safe pilings" of cubes in the corner of a room: the height should not increase away from the corner - Wouter Meeussen (w.meeussen.vdmcc@vandemoortele.be). %D A000219 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241. %D A000219 A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. %D A000219 Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 18, Feb. 1972. %D A000219 Bender, E. A. and Knuth, D. E. Enumeration of Plane Partitions.'' J. Combin. Theory A. 13, 40-54, 1972. %D A000219 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; pp(n) on p. 10. %D A000219 D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646. %D A000219 L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.6). %D A000219 D. E. Knuth, A Note on Solid Partitions. Math. Comp. 24, 955-961, 1970. %D A000219 P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332. %H A000219 Beeler, M., Gosper, R.W., and Schroeppel, R., HAKMEM, ITEM 18 %H A000219 M. Beeler, R. W. Gosper and R. Schroeppel, HAKMEM (Web version due to Henry Baker) %H A000219 H. Bottomley, Illustration of initial terms %H A000219 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 137 %H A000219 E. W. Weisstein, Link to a section of The World of Mathematics. %H A000219 Index entries for "core" sequences %F A000219 G.f.: prod from k = 1 to inf ( 1 - x^k )^(-k). %e A000219 A planar partition of 13: %e A000219 4 3 1 1 %e A000219 2 1 %e A000219 1 %p A000219 series(mul((1-x^k)^(-k),k=1..64),x,63); %t A000219 Rest@CoefficientList[ Series[ Product[ (1-x^k)^-k,{k,1,64} ],{x,0,64} ],x ] %Y A000219 Cf. A000784, A000785, A000786, A005987, A048141, A048142. %K A000219 nonn,nice,easy,core %O A000219 0,3 %A A000219 njas %I A027999 %S A027999 1,0,1,3,6,13,24,49,91,181,334,632,1163,2138,3880,7006, %T A027999 12531,22279,39369,69078,120597,209282,361405,620829,1061687, %U A027999 1807014,3062642,5168784,8688820,14549659,24274226,40353748 %N A027999 Expansion of Product(1+q^m)^(m(m-1)/2); m=1..inf. %K A027999 nonn %O A027999 0,4 %A A027999 njas %I A005196 M2567 %S A005196 1,3,6,13,24,49,93,190,381,803,1703,3755,8401,19338,45275,108229, %T A005196 262604,647083,1613941,4072198,10374138,26663390,69056163,180098668 %N A005196 Random forests. %D A005196 E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121. %K A005196 nonn,nice %O A005196 1,2 %A A005196 njas %I A032287 %S A032287 1,3,6,13,24,51,97,207,428,946,2088,4831,11209,26717,64058,155725, %T A032287 380400,936575,2314105,5744700,14300416,35708268,89359536, %U A032287 224121973,563126689,1417378191,3572884062,9019324297,22797540648 %N A032287 "DIK" (bracelet, indistinct, unlabeled) transform of 1,2,3,4... %H A032287 Index entries for sequences related to bracelets %H A032287 C. G. Bower, Transforms (2) %K A032287 nonn %O A032287 1,2 %A A032287 Christian G. Bower (bowerc@usa.net) %I A006017 M2568 %S A006017 1,3,6,13,24,52,103,222,384,832,1648,3552,6237,13563,26511, %T A006017 56906,98304,212992,421888,909312,1596672,3472128,6786816,14567936, %U A006017 25190110,54589881,108036850,232800673,408783316,888883132 %N A006017 Nim product 2^n . 2^n. %D A006017 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 444. %H A006017 Index entries for sequences related to Nim-multiplication %K A006017 nonn %O A006017 0,2 %A A006017 njas,jos %I A047183 %S A047183 0,0,0,3,6,13,25,45,91,154,322,525,1125,1793,3921,6153,13671,21206, %T A047183 47718,73359,166800,254607,583947,886171,2047375,3091971,7188451, %U A047183 10811671,25272481,37877401,88959929,132924649,313498639 %N A047183 Nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= (n-2)/2. %K A047183 nonn %O A047183 1,4 %A A047183 Clark Kimberling, ck6@cedar.evansville.edu %I A047194 %S A047194 0,0,1,3,6,13,25,45,91,175,322,645,1245,2325,4651,9031,17061,34123, %T A047194 66547,126763,253527,496063,950818,1901637,3730293,7184421,14368843, %U A047194 28243063,54604081,109208163,215008363,416990563,833981127 %N A047194 Nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= n/3. %K A047194 nonn %O A047194 1,4 %A A047194 Clark Kimberling, ck6@cedar.evansville.edu %I A048039 %S A048039 0,0,1,3,6,13,25,45,91,175,322,645,1245,2325,4651,9031,17061,34123, %T A048039 66547,126763,253527,496063,950818,1901637,3730293,7184421,14368843, %U A048039 28243063,54604081,109208163,215008363,416990563,833981127 %N A048039 Number of nonempty subsets of {1,2,...,n} in which exactly 1/3 of the elements are <= (n+1)/3. %K A048039 nonn %O A048039 1,4 %A A048039 Clark Kimberling, ck6@cedar.evansville.edu %I A005406 M2569 %S A005406 1,3,6,13,25,49,91,170,309,558 %N A005406 Protruded partitions of n. %D A005406 R. P. Stanley, A Fibonacci lattice, Fib. Quart., 13 (1975), 215-232. %K A005406 nonn %O A005406 1,2 %A A005406 njas %I A005407 M2570 %S A005407 1,3,6,13,25,50,93,175,320,582 %N A005407 Protruded partitions of n. %D A005407 R. P. Stanley, A Fibonacci lattice, Fib. Quart., 13 (1975), 215-232. %K A005407 nonn %O A005407 1,2 %A A005407 njas %I A005116 M2571 %S A005116 1,3,6,13,25,50,94,178,328,601 %N A005116 Protruded partitions of n. %D A005116 R. P. Stanley, A Fibonacci lattice, Fib. Quart., 13 (1975), 215-232. %K A005116 nonn %O A005116 1,2 %A A005116 njas, rps %I A032198 %S A032198 1,3,6,13,25,58,121,283,646,1527,3601,8678,20881,50823,124054, %T A032198 304573,750121,1855098,4600201,11442085,28527446,71292603, %U A032198 178526881,447919418,1125750145,2833906683,7144450566,18036423973 %N A032198 "CIK" (necklace, indistinct, unlabeled) transform of 1,2,3,4... %H A032198 Index entries for sequences related to necklaces %H A032198 C. G. Bower, Transforms (2) %K A032198 nonn %O A032198 1,2 %A A032198 Christian G. Bower (bowerc@usa.net) %I A019300 %S A019300 0,1,3,6,13,26,52,105,211,422,844,1689,3378,6757,13515,27030,54061, %T A019300 108122,216244,432489,864978,1729957,3459915,6919830,13839660,27679321, %U A019300 55358643,110717286,221434573,442869146,885738292,1771476585,3542953171 %N A019300 First n elements of Thue-Morse sequence A010060 read as a binary number. %K A019300 nonn %O A019300 0,3 %A A019300 Jonas Wallgren (jwc@ida.liu.se) %I A033129 %S A033129 1,3,6,13,27,54,109,219,438,877,1755,3510,7021,14043,28086,56173, %T A033129 112347,224694,449389,898779,1797558,3595117,7190235,14380470, %U A033129 28760941,57521883,115043766,230087533,460175067,920350134,1840700269 %N A033129 Base 2 digits are, in order, the first n terms of the periodic sequence with initial period 1,1,0. %K A033129 nonn,base %O A033129 1,2 %A A033129 Clark Kimberling, ck6@cedar.evansville.edu %I A065830 %S A065830 1,3,6,13,27,55,110,220,440,881,1762,3524,7048,14097,28195,56390, %T A065830 112781,225562,451125,902250,1804500,3609000,7218000,14436001,28872002, %U A065830 57744005,115488011,230976023,461952047,923904094,1847808189 %N A065830 A065829 converted to base 10. %H A065830 G. L. Honaker, Jr. and C. K. Caldwell, Prime Curios! 881 %Y A065830 Cf. A065828 up to A065840, A000796, A011545, A011546, A055143. %K A065830 nonn,base %O A065830 1,2 %A A065830 Patrick De Geest (pdg@worldofnumbers.com), Nov 24 2001. %I A055143 %S A055143 1,3,6,13,27,55,110,220,441,882,1765,3531,7063,14126,28253,56507, %T A055143 113015,226031,452062,904124,1808248,3616497,7232994,14465988,28931977, %U A055143 57863955,115727910,231455821,462911642,925823285,1851646570 %N A055143 The first n digits of the juxtaposition of the base 2 numbers converted to decimal. %e A055143 1 (1); 11 (3); 110 (6); 1101 (13); 11011 (27); 110111 (55); ... %Y A055143 Cf. A030190, A054633, A055074, A055144-A055150. %K A055143 nonn,base %O A055143 1,2 %A A055143 Patrick De Geest (pdg@worldofnumbers.com), Apr 2000. %I A036886 %S A036886 1,3,6,13,27,61,132,285,590 %N A036886 Partitions of 5n satisfying either one of the two conditions cn(0,5) = cn(1,5) = cn(4,5) < cn(2,5) = cn(3,5) or cn(0,5) = cn(2,5) = cn(3,5) < cn(1,5) = cn(4,5). %C A036886 For a given partition cn(i,n) means the number of its parts equal to i modulo n. %C A036886 Short: (0 = 1 = 4 < 2 = 3) or (0 = 2 = 3 < 1 = 4). %Y A036886 Extracted from A036836. %K A036886 nonn,more,part %O A036886 1,2 %A A036886 Olivier Gerard (ogerard@ext.jussieu.fr) %I A052251 %S A052251 1,3,6,13,27,63,148,363,912,2330,6036 %N A052251 Column 3 of A052250. %K A052251 nonn %O A052251 3,2 %A A052251 D.Broadhurst@open.ac.uk (David Broadhurst), Feb 05 2000 %I A032253 %S A032253 3,6,13,27,78,278,1011,3753,13843,50934,187629,692891,2568882, %T A032253 9562074,35742329,134117829,505093740,1908474674,7232842785, %U A032253 27486193251,104712247296,399816026490,1529742725403 %N A032253 "DHK" (bracelet, identity, unlabeled) transform of 3,3,3,3... %H A032253 C. G. Bower, Transforms (2) %K A032253 nonn %O A032253 1,1 %A A032253 Christian G. Bower (bowerc@usa.net) %I A002478 M2572 N1017 %S A002478 1,1,3,6,13,28,60,129,277,595,1278,2745,5896,12664,27201,58425,125491, %T A002478 269542,578949,1243524,2670964,5736961,12322413,26467299,56849086, %U A002478 122106097,262271568,563332848,1209982081,2598919345,5582216355 %N A002478 Bisection of A000930. %C A002478 Number of ways to tile a 3 X n region with 1 X 1, 2 X 2 and 3 X 3 tiles. %D A002478 L. Euler, Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 322. %D A002478 S. Heubach, Tiling an m-by-n Area with Squares of Size up to k-by-k (m<=5), Congressus Numerantium 140 (1999), pp. 43-64. %H A002478 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 412 %F A002478 a(n)=a(n-1)+2a(n-2)+a(n-3) %e A002478 a(3)=6 as there is one tiling of a 3x3 region with only 1 X 1 tiles, 4 tilings with exactly one 2 X 2 tile and 1 tiling consisting of the 3 X 3 tile. %t A002478 f[ A_ ] := Module[ {til = A},AppendTo[ til, A[ [ -1 ] ] + 2A[ [ -2 ] ] + A[ [ -3 ] ] ] ]; NumOfTilings[ n_ ] := Nest[ f, {1, 1, 3}, n - 2 ]; NumOfTilings[ 30 ] %Y A002478 Cf. A054856, A054857. %K A002478 easy,nonn,nice %O A002478 0,3 %A A002478 njas %E A002478 Additional comments from Silvia Heubach (silvi@cine.net), Apr 21 2000 %I A052933 %S A052933 1,1,3,6,13,29,62,137,297,650,1417,3093,6750,14729,32145,70146,153081, %T A052933 334061,729014,1590905,3471785,7576378,16533705,36081029,78738574, %U A052933 171828905,374977217,818301874,1785756377,3897004189,8504318886 %N A052933 A simple regular expression. %H A052933 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 920 %F A052933 G.f.: -(-1+x^2)/(1-3*x^2+2*x^4-x) %F A052933 Recurrence: {a(1)=1,a(0)=1,a(3)=6,a(2)=3,2*a(n)-3*a(n+2)-a(n+3)+a(n+4)} %F A052933 Sum(-1/397*(-148*_alpha+27*_alpha^2+82*_alpha^3-51)*_alpha^(-1-n),_alpha=RootOf(1-3*_Z^2+2*_Z^4-_Z)) %p A052933 spec:= [S,{S=Sequence(Prod(Z,Union(Z,Z,Sequence(Prod(Z,Z)))))},unlabelled ]: seq(combstruct[count ](spec,size=n),n=0..20); %K A052933 easy,nonn %O A052933 0,3 %A A052933 encyclopedia@pommard.inria.fr, Jan 25 2000 %E A052933 More terms from James A. Sellers (sellersj@math.psu.edu), Jun 06 2000 %I A018909 %S A018909 3,6,13,29,65,146,328,737,1657,3726,8379,18843,42375,95295,214305, %T A018909 481942,1083821,2437364,5481296,12326680,27721007,62340730, %U A018909 140195723,315280889,709023335,1594495915,3585801902,8063975053 %N A018909 Define the sequence S(a_0,a_1) by a_{n+2} is the least integer such that a_{n+2}/a_{n+1}>a_{n+1}/a_n for n >= 0. This is S(3,6). %D A018909 D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993;. %K A018909 nonn %O A018909 0,1 %A A018909 rkg@cpsc.ucalgary.ca (Richard Guy) %I A005313 M2573 %S A005313 1,3,6,13,29,70,175,449,1164,3035,7931,20748,54301,142143,372114,974185, %T A005313 2550425,6677074,17480779,45765245,119814936,313679543,821223671,2149991448 %N A005313 Triangular anti-Hadamard matrices of order n. %D A005313 R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137. %K A005313 nonn %O A005313 1,2 %A A005313 njas %I A018014 %S A018014 1,3,6,13,31,72,169,398,935,2197,5166,12147,28561,67157, %T A018014 157908,371293,873035,2052796,4826809,11349444,26686341, %U A018014 62748517,147542765,346922421,815730721,1918055941,4509991466 %N A018014 Powers of cube root of 13 rounded up. %K A018014 nonn %O A018014 0,2 %A A018014 njas %I A026538 %S A026538 1,1,3,6,13,33,65,180,346,990,1897,5502,10571,30863,59523,174456, %T A026538 337672,992304,1926650,5673140,11043858,32571858,63548069,187675644, %U A026538 366849016,1084649644,2123604927,6284986554,12322549765,36501029265 %N A026538 a(n)=T(n,n-1), T given by A026536. Also a(n) = number of integer strings s(0),...,s(n), counted by T, such that s(n)=1. %K A026538 nonn %O A026538 1,3 %A A026538 Clark Kimberling, ck6@cedar.evansville.edu %I A062466 %S A062466 0,1,1,3,6,13,34,91,257,754,2290,7185,23216,77019,261740,909353, %T A062466 3224262,11649471,42833407,160084445,607507992,2338760045,9126115359, %U A062466 36068187762,144279012291,583783534411,2387939881857,9869420688876 %N A062466 Nearest integer to ln(n)^(n^(1 - 1/n)). %t A062466 Round[Log[n]^n^(1 - 1/n)] %Y A062466 Cf. A062465. %K A062466 nonn,easy %O A062466 1,4 %A A062466 Olivier Gerard (ogerard@ext.jussieu.fr), Jun 23 2001 %I A053564 %S A053564 0,0,1,3,6,13,36,93,243,645,1782,4914,13608,37890,106288,299025,844182, %T A053564 2391363,6797196,19371684,55345784,158486625,454795398,1307541690, %U A053564 3765720066,10862647236,31381059609,90780960426,262951692390 %N A053564 Number of ternary Lyndon words with trace 1 and subtrace 2. %C A053564 Same as number of ternary Lyndon words with trace 2 and subtrace 2. %H A053564 F. Ruskey, Ternary Lyndon words with give trace and subtrace %F A053564 (1/n) SUM mobius(d) M(n/d,1,2); d|n, d=1,2(3) where M(n,t,s) = SUM n!/(i!j!k!); i+j+k=n, j=t(3), k=s(3). %e A053564 a(4) = 3 = |{ 0112, 0121, 0211 }| %K A053564 nonn %O A053564 1,4 %A A053564 Frank Ruskey (fruskey@csr.csc.uvic.ca), Jan 17 2000 %I A036781 %S A036781 1,3,6,13,38,159,880,5921,46242,409123,4037924,43954725,522956326, %T A036781 6749977127,93928268328,1401602636329,22324392524330,378011820620331, %U A036781 6780385526348332,128425485935180333,2561327494111820334 %N A036781 n + Sum_{k=0..n} k!. %Y A036781 A003422[ n ] + n. %K A036781 nonn %O A036781 0,2 %A A036781 njas %I A055738 %S A055738 0,3,6,13,39,167,900,4769,28389 %N A055738 Number of primes quadruplets < 10^n, where a prime quadruplet means 4 successive primes {p, p', p'', p'''} with p''' = p + 8. %D A055738 J. Recreational Math., vol. 23, No. 2, 1991, p. 97. %e A055738 For n=2 the quadruplets are 3,5,7,11; 5,7,11,13; 11,13,17,19. %p A055738 with(numtheory): x:=1229; t1:=[seq(ithprime(i),i=1..x)]; c:=0: for i from 1 to x-3 do if t1[i]+8 = t1[i+3] then c:=c+1; fi; od: c; # the values of x to use are given by A006880. %t A055738 x=168; a=Table[ Prime[ n ],{n,1,x} ]; c=0; Do[ If[ a[ [ n ] ]+8==a[ [ n+2 ] ],c++ ],{n,1,x-3} ]; # the values of x to use are given by A006880. %Y A055738 Cf. A055737, A006880. %K A055738 more,nonn %O A055738 1,2 %A A055738 Robert G. Wilson v (rgwv@kspaint.com), Jun 09 2000 %E A055738 2 more terms from Jud McCranie (jud.mccranie@mindspring.com), Oct 08 2000. %I A053365 %S A053365 3,6,14,15,20,27,38,123,276,327,380,411,546,819,1155,1274,1800,1875, %T A053365 3135,3411,7514,9446,12615,23400,23564,99344,108258 %N A053365 297*2^n+1 is prime. %H A053365 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %H A053365 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300 %H A053365 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300 %K A053365 hard,nonn %O A053365 0,1 %A A053365 njas, Dec 29 1999 %I A056596 %S A056596 0,1,3,6,14,24,54,88,148,240,510,756,1548,2520,3936,5248,10624,14508, %T A056596 29196,40740,60500,95400,191400,242016,338880,529920,674688,912912, %U A056596 1830192,2327424,4660224,5523456,7858176,12152064,16406592,19576080 %N A056596 Number of nonsquare divisors of n!. %F A056596 a(n)=d(n!)-A046951(n!) %Y A056596 A000142, A000005, A046951, A055772. %K A056596 nonn %O A056596 1,3 %A A056596 Labos E. (labos@ana1.sote.hu), Jul 21 2000 %I A026341 %S A026341 0,3,6,14,25,24,38,48,75,89,114,142,141,160,193,229,228,267,293, %T A026341 354,402,434,487,486,602,601,663,728,771,770,840,886,961,1039, %U A026341 1038,1204,1203,1258,1257,1439,1438,1533,1595,1695,1836,1904 %N A026341 a(n) = sum of the numbers between the two n's in A026338. %K A026341 nonn %O A026341 1,2 %A A026341 Clark Kimberling, ck6@cedar.evansville.edu %I A026271 %S A026271 0,3,6,14,25,33,49,60,81,105,121,150,182,203,240,264,306,351, %T A026271 380,430,462,517,575,612,675,741,783,854,899,975,1054,1104,1188, %U A026271 1275,1330,1422,1480,1577,1677,1740,1845,1911,2021,2134,2205 %N A026271 a(n) = sum of the numbers between the two n's in A026242. %K A026271 nonn %O A026271 1,2 %A A026271 Clark Kimberling, ck6@cedar.evansville.edu %I A002219 M2574 N1018 %S A002219 1,3,6,14,25,53,89,167,278,480,760,1273,1948,3089,4682,7177,10565, %T A002219 15869,22911,33601,47942,68756,96570,136883,189674,264297,362995, %U A002219 499617,678245,924522,1243098,1676339,2237625,2988351,3957525,5247500 %N A002219 Number of cycle types of sums of two degree n permutations, %C A002219 Also biquanimous partitions of 2n. (A biquanimous partition is one that can be bisected into two equal sized parts: e.g. 3+2+1 is a biquanimous partition of 6 as it contains 3 and 2+1, but 5+1 is not.) %D A002219 N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376. %Y A002219 Cf. A064914. %K A002219 nonn,nice %O A002219 1,2 %A A002219 njas %E A002219 Better description from Vladeta Jovovic (vladeta@Eunet.yu), Mar 06 2000 %E A002219 More terms and additional comments by Christian G. Bower (bowerc@usa.net), Oct 12 2001 %I A006906 M2575 %S A006906 1,1,3,6,14,25,56,97,198,354,672,1170,2207,3762,6786,11675,20524,34636, %T A006906 60258,100580,171894,285820,480497,791316,1321346,2156830,3557353, %U A006906 5783660,9452658,15250216,24771526,39713788,64011924,102199026 %N A006906 Sum of products of terms in all partitions of n. %D A006906 G. Labelle, personal communication. %F A006906 G.f.: 1 / Product (1-kx^k). %e A006906 The partitions of 4 are 4, 1+3, 2+2, 2+1+1, 1+1+1+1, the corresponding products are 4,3,4,2,1, and their sum is a(4) = 14. %Y A006906 Cf. A007870. %K A006906 nonn,nice,easy %O A006906 0,3 %A A006906 sp %E A006906 More terms from Vladeta Jovovic (vladeta@Eunet.yu), Oct 04 2001 %I A049940 %S A049940 1,1,3,6,14,26,54,119,278,503,1008,2027,4094,8412,17554,38194,89848, %T A049940 162143,324288,648587,1297214,2594652,5190034,10383154,20779768, %U A049940 41631830,83498100,167969126,339831072,695251878,1453222088 %N A049940 a(n)=a(1)+a(2)+...+a(n-1)+a(m), where m=2n-3-2^(p+1), and 2^p 0. %Y A051749 Cf. A006907, A051748. %K A051749 nonn,nice %O A051749 1,2 %A A051749 JOHN MCKAY (mac@discrete.concordia.ca), Dec 08 1999 %I A030012 %S A030012 1,3,6,14,27,58,109,216,402,760,1390,2550,4569,8178,14408, %T A030012 25280,43850,75685,129436,220226,371906,624840,1043178, %U A030012 1733108,2863422,4709222,7706800,12558213,20372860,32917707 %N A030012 Euler transform of {1,primes}. %H A030012 N. J. A. Sloane, Transforms %K A030012 nonn %O A030012 0,2 %A A030012 njas (end) Sloane's Database of Integer Sequences, Part 27 Part of the On-Line Encyclopedia of Integer Sequences This is a section of the main database for the On-Line Encyclopedia of Integer Sequences. For more information see the following pages: ( www.research.att.com/~njas/sequences/ then ) Seis.html: Welcome index.html: Lookup indexfr.html: Francais demo1.html: Demos Sindx.html: Index WebCam.html: WebCam Submit.html: Contribute new sequence or comment eishelp1.html: Internal format eishelp2.html: Beautified format transforms.html: Transforms Spuzzle.html: Puzzles Shot.html: Hot classic.html: Classics ol.html: Superseeker JIS/index.html: Journal of Integer Sequences pages.html: More pages Maintained by: N. J. A. Sloane (njas@research.att.com), home page: www.research.att.com/~njas/