Sloane's Database of Integer Sequences, Part 55
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 A004315
%S A004315 1,20,231,2024,14950,98280,593775,3365856,18156204,94143280,
%T A004315 472733756,2311801440,11058116888,51915526432,239877544005,
%U A004315 1093260079344,4923689695575,21945588357420,96926348578605
%N A004315 Binomial coefficient C(2n,n-9).
%D A004315 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.
%K A004315 nonn,easy
%O A004315 9,2
%A A004315 njas
%I A061139
%S A061139 0,0,0,0,0,20,240,1260,5600,45360,383040,2451680,17128320,157769040,
%T A061139 1902380480,18882623760,163633317120,2095059774080,30792478993920,
%U A061139 346562329685760,3905491275514880,58609449249207360,866031730098205440
%N A061139 Degree n odd permutations of order exactly 6.
%F A061139 E.g.f.: - 1/2*exp(x + 1/2*x^2) + 1/2*exp(x - 1/2*x^2) + 1/2*exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) - 1/2*exp(x - 1/2*x^2 + 1/3*x^3 - 1/6*x^6).
%Y A061139 Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135, A001465, A061136 - A061140.
%K A061139 easy,nonn
%O A061139 0,6
%A A061139 Vladeta Jovovic (vladeta@Eunet.yu), Apr 14 2001
%I A061121
%S A061121 0,0,0,0,20,240,1470,10640,83160,584640,4496030,42658440,371762820,
%T A061121 3594871280,38650622010,396457108320,4330689250160,53963701424640,
%U A061121 641211774798510,8205894865096280,113786291585124060
%N A061121 Degree n permutations of order exactly 6.
%F A061121 E.g.f.: exp(x)-exp(x+1/2*x^2)-exp(x+1/3*x^3)+exp(x+1/2*x^2+1/3*x^3+1/6*x^6).
%Y A061121 Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.
%K A061121 easy,nonn
%O A061121 1,5
%A A061121 Vladeta Jovovic (vladeta@Eunet.yu), Apr 14 2001
%I A040075
%S A040075 1,20,240,2240,17920,129024,860160,5406720,32440320,187432960,
%T A040075 1049624576,5725224960,30534533120,159719096320,821412495360,
%U A040075 4161823309824,20809116549120,102821517066240,502682972323840
%N A040075 5-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^5.
%C A040075 Also convolution of A020920 with A000984 (central binomial coefficients).
%H A040075 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A040075 a(n)=binomial(n+4,4)*4^n; G.f. 1/(1-4*x)^5.
%Y A040075 Cf. A000302, A020920, A000984.
%K A040075 easy,nonn
%O A040075 0,2
%A A040075 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A055757
%S A055757 1,0,0,20,246,0,0,30624,127908,0,0,3699612,9190616,0,0,95498592,
%T A055757 188170398,0,0,1143506364,1960018920,0,0,8506347552,13291928232,0,0,
%U A055757 45759995408,67075871808,0,0,195397296192,272568671892,0,0
%N A055757 Jacobi form of weight 12 and index 1 for Niemeier lattice of type E_6^4 or A_11 D_7 E_6.
%D A055757 Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser,1985.
%F A055757 E_8*E_{4,1}-36*phi_12
%Y A055757 A008695.
%K A055757 nonn
%O A055757 0,4
%A A055757 Kok Seng Chua (chuaks@ihpc.nus.edu.sg), Jul 12 2000
%I A022744
%S A022744 1,20,250,2400,19375,137604,884430,5241000,29017815,151597440,
%T A022744 752997538,3577442200,16335561280,71979549460,307075300540,
%U A022744 1271938667704,5127690095315,20161295885120,77454572685460
%N A022744 Expansion of Product (1-m*q^m)^-20; m=1..inf.
%K A022744 nonn
%O A022744 0,2
%A A022744 njas
%I A028032
%S A028032 1,20,255,2650,24521,210840,1725235,13631750,104995341,
%T A028032 793395460,5908353815,43502236050,317443722961,2299938136880,
%U A028032 16567618807995,118785894675550,848393691575381,6040178746163100
%N A028032 Expansion of 1/((1-3x)(1-4x)(1-6x)(1-7x)).
%K A028032 nonn
%O A028032 0,2
%A A028032 njas
%I A025986
%S A025986 1,20,257,2704,25389,221676,1841449,14758568,115171397,
%T A025986 880623172,6627177921,49248518592,362296167325,2643446894108,
%U A025986 19158543110873,138087153497176,990728497913973,7081081591668084
%N A025986 Expansion of 1/((1-2x)(1-5x)(1-6x)(1-7x)).
%K A025986 nonn
%O A025986 0,2
%A A025986 njas
%I A028027
%S A028027 1,20,257,2716,25809,230244,1975009,16524332,136058417,
%T A028027 1108775668,8975764161,72350153148,581586939025,4666887733892,
%U A028027 37407122372513,299621333407564,2398809490126833,19199738367402516
%N A028027 Expansion of 1/((1-3x)(1-4x)(1-5x)(1-8x)).
%K A028027 nonn
%O A028027 0,2
%A A028027 njas
%I A025966
%S A025966 1,20,260,2800,27216,248640,2182720,18656000,156544256,
%T A025966 1296655360,10641146880,86744985600,703688298496,5688011079680,
%U A025966 45855653642240,368956766617600,2964331947687936,23790756829593600
%N A025966 Expansion of 1/((1-2x)(1-4x)(1-6x)(1-8x)).
%K A025966 nonn
%O A025966 0,2
%A A025966 njas
%I A041764
%S A041764 20,261,542,803,2951,3754,14213,17967,50147,669878,26845267,
%T A041764 349658349,726161965,1075820314,3953622907,5029443221,19041952570,
%U A041764 24071395791,67184744152,897473069767,35966107534832,468456871022583
%N A041764 Numerators of continued fraction convergents to sqrt(403).
%Y A041764 Cf. A041765.
%K A041764 nonn,cofr,easy
%O A041764 0,1
%A A041764 njas
%I A022111
%S A022111 1,20,263,2878,28449,264048,2350651,20332466,172311557,
%T A022111 1438844836,11885079999,97387603014,793247778025,6432389826584,
%U A022111 51985193621507,419076145997722,3371967484999053,27092843456412492
%N A022111 Expansion of 1/((1-x)(1-5x)(1-6x)(1-8x)).
%K A022111 nonn
%O A022111 0,2
%A A022111 njas
%I A025943
%S A025943 1,20,263,2890,28833,271320,2457211,21670550,187473605,
%T A025943 1598611300,13480822719,112690635330,935435043817,7720854845360,
%U A025943 63428474542787,519059289782830,4233894901355469,34441103679651900
%N A025943 Expansion of 1/((1-2x)(1-3x)(1-7x)(1-8x)).
%K A025943 nonn
%O A025943 0,2
%A A025943 njas
%I A025961
%S A025961 1,20,263,2902,29289,281472,2630731,24196634,220500797,
%T A025961 1998686404,18061169919,162923877246,1468215185425,13223572329416,
%U A025961 119060866122227,1071793518338338,9647378871105573,86832630229860108
%N A025961 Expansion of 1/((1-2x)(1-4x)(1-5x)(1-9x)).
%K A025961 nonn
%O A025961 0,2
%A A025961 njas
%I A021854
%S A021854 1,20,265,2940,29601,280740,2558905,22683980,197048401,
%T A021854 1685772660,14253277545,119401486620,992861126401,8206523123780,
%U A021854 67497981692185,552905862106860,4513706683463601,36742894880462100
%N A021854 Expansion of 1/((1-x)(1-4x)(1-7x)(1-8x)).
%K A021854 nonn
%O A021854 0,2
%A A021854 njas
%I A025939
%S A025939 1,20,265,2960,30301,295340,2795905,25996520,238988101,
%T A025939 2181067460,19810829545,179385327680,1620996705901,14628147660380,
%U A025939 131888407103185,1188406175868440,10704118783463701,96387848642218100
%N A025939 Expansion of 1/((1-2x)(1-3x)(1-6x)(1-9x)).
%K A025939 nonn
%O A025939 0,2
%A A025939 njas
%I A021824
%S A021824 1,20,267,3010,31073,304920,2901319,27075950,249555405,
%T A021824 2281579300,20749095731,188036746170,1700122392697,15347941075760,
%U A021824 138412864844703,1247405586235270,11236794818538149,101192043527859900
%N A021824 Expansion of 1/((1-x)(1-4x)(1-6x)(1-9x)).
%K A021824 nonn
%O A021824 0,2
%A A021824 njas
%I A019928
%S A019928 1,20,269,3040,31161,300300,2775109,24887960,218303921,
%T A019928 1882786180,16026538749,135010883280,1127921219881,9359429537660,
%U A019928 77233958267189,634411837477000,5191228487083041,42342127346986740
%N A019928 Expansion of 1/((1-5x)(1-7x)(1-8x)).
%K A019928 nonn
%O A019928 0,2
%A A019928 njas
%I A025934
%S A025934 1,20,269,3100,33261,344580,3507709,35392940,355527821,
%T A025934 3563328340,35673709149,356939747580,3570412342381,35709202523300,
%U A025934 357117435046589,3571301442515020,35713650014404941,357139678477481460
%N A025934 Expansion of 1/((1-2x)(1-3x)(1-5x)(1-10x)).
%K A025934 nonn
%O A025934 0,2
%A A025934 njas
%I A021594
%S A021594 1,20,270,3100,32711,328440,3195340,30437000,285695421,
%T A021594 2653625260,24459281210,224170373700,2045792060131,18609941810480,
%U A021594 168874176353880,1529560509125200,13833895100278841,124980009723284100
%N A021594 Expansion of 1/((1-x)(1-3x)(1-7x)(1-9x)).
%K A021594 nonn
%O A021594 0,2
%A A021594 njas
%I A019793
%S A019793 1,20,271,3110,32641,324800,3125011,29414090,272851381,
%T A019793 2506362980,22871235751,207773763470,1881803862121,17008495407560,
%U A019793 153516126074491,1384313656687250,12474986630176861,112372624930994540
%N A019793 Expansion of 1/((1-5x)(1-6x)(1-9x)).
%K A019793 nonn
%O A019793 0,2
%A A019793 njas
%I A021774
%S A021774 1,20,271,3150,34041,354480,3620611,36607010,368161981,
%T A021774 3692428740,36979730151,370080107670,3702237477121,37029646251800,
%U A021774 370333177834891,3703516786589130,37036098633715461,370365663082767660
%N A021774 Expansion of 1/((1-x)(1-4x)(1-5x)(1-10x)).
%K A021774 nonn
%O A021774 0,2
%A A021774 njas
%I A019613
%S A019613 1,20,273,3172,33809,342132,3348241,32033924,301669137,
%T A019613 2808831124,25937190929,238042888356,2174659962385,19797924540596,
%U A019613 179763483454737,1628947562960068,14738065844679953,133185374228264148
%N A019613 Expansion of 1/((1-4x)(1-7x)(1-9x)).
%K A019613 nonn
%O A019613 0,2
%A A019613 njas
%I A021514
%S A021514 1,20,273,3208,35069,368988,3800761,38676176,390782997,
%T A021514 3931986916,39464899409,395519441304,3960417893485,39635522209004,
%U A021514 396543288909417,3966561311533792,39672383714545733,396764460934414452
%N A021514 Expansion of 1/((1-x)(1-3x)(1-6x)(1-10x)).
%K A021514 nonn
%O A021514 0,2
%A A021514 njas
%I A021264
%S A021264 1,20,275,3250,35481,369240,3722575,36698750,355853861,
%T A021264 3407206660,32301037275,303798758250,2838904214641,26387861071280,
%U A021264 244192534790375,2251347094369750,20691038099509821,189650656897307100
%N A021264 Expansion of 1/((1-x)(1-2x)(1-8x)(1-9x)).
%K A021264 nonn
%O A021264 0,2
%A A021264 njas
%I A025928
%S A025928 1,20,275,3310,37761,421440,4662175,51395570,565817021,
%T A025928 6225908260,68492850075,753453315030,8288115908281,91169797529480,
%U A025928 1002869877293975,11031577111093690,121347382194479541
%N A025928 Expansion of 1/((1-2x)(1-3x)(1-4x)(1-11x)).
%K A025928 nonn
%O A025928 0,2
%A A025928 njas
%I A004334
%S A004334 1,20,276,3276,35960,376992,3838380,38320568,377348994,
%T A004334 3679075400,35607051480,342700125300,3284214703056,31368725759168,
%U A004334 298824321028320,2840671544105280,26958221130508525,255485622301674660
%N A004334 Binomial coefficient C(4n,n-4).
%D A004334 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.
%K A004334 nonn,easy
%O A004334 4,2
%A A004334 njas
%I A019483
%S A019483 1,20,276,3280,36176,383040,3962176,40428800,409195776,
%T A019483 4121666560,41395966976,415039672320,4156893515776,41607983022080,
%U A019483 416314385842176,4164552265891840,41653977398706176,416590519605657600
%N A019483 Expansion of 1/((1-4x)(1-6x)(1-10x)).
%K A019483 nonn
%O A019483 0,2
%A A019483 njas
%I A018056
%S A018056 1,20,277,3296,36169,377804,3819229,37727192,366384337,
%T A018056 3512195588,33327711781,313693195088,2933189599705,27278314742972,
%U A018056 252541704234733,2329170324845384,21412892860517473,196318915369069556
%N A018056 Expansion of 1/((1-3x)(1-8x)(1-9x)).
%K A018056 nonn
%O A018056 0,2
%A A018056 njas
%I A021234
%S A021234 1,20,277,3324,37149,398916,4181269,43157708,440992717,
%T A021234 4475837652,45219751941,455427151452,4576878947005,45927041513828,
%U A021234 460378179477493,4611536145214956,46169641905360813,462076382226349044
%N A021234 Expansion of 1/((1-x)(1-2x)(1-7x)(1-10x)).
%K A021234 nonn
%O A021234 0,2
%A A021234 njas
%I A021474
%S A021474 1,20,278,3388,39039,438648,4872316,53834696,593387597,
%T A021474 6533322796,71896935474,791018479524,8701965018475,95725426313264,
%U A021474 1052998752170552,11583081609022672,127414374439552473
%N A021474 Expansion of 1/((1-x)(1-3x)(1-5x)(1-11x)).
%K A021474 nonn
%O A021474 0,2
%A A021474 njas
%I A017999
%S A017999 1,20,279,3370,37841,407640,4281739,44256950,452652981,
%T A017999 4597133860,46465625999,468116448930,4705386343321,47223418005680,
%U A017999 473421066847059,4742518890351310,47483346499724861,475240568299871100
%N A017999 Expansion of 1/((1-3x)(1-7x)(1-10x)).
%K A017999 nonn
%O A017999 0,2
%A A017999 njas
%I A012836
%S A012836 1,0,20,280,18000,1277760,159922880,26119949440,5923118588160,
%T A012836 1705199348152320,619341836139299840,274598298613483878400,
%U A012836 146431095418622106193920,92305017354779154307399680
%V A012836 1,0,20,-280,18000,-1277760,159922880,-26119949440,5923118588160,
%W A012836 -1705199348152320,619341836139299840,-274598298613483878400,
%X A012836 146431095418622106193920,-92305017354779154307399680
%N A012836 arcsinh(sec(x)*tanh(x))=x+20/5!*x^5-280/7!*x^7+18000/9!*x^9...
%K A012836 sign,done
%O A012836 0,3
%A A012836 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A028294
%S A028294 1,20,281,1357,4281,10666,22825,43891,77937,130096,
%T A028294 206681,315305,465001,666342,931561,1274671
%N A028294 Number of stacks of n pikelets, distance 5 flips from a well-ordered stack.
%D A028294 Marc Paulhus (paulhusm@math.ucalgary.ca), Pikelets, Discrete Math, to be submitted.
%F A028294 n^5 - 65/6*n^4 + 173/6*n^3 + 148/3*n^2 - 862/3*n + 265.
%K A028294 nonn
%O A028294 4,2
%A A028294 Richard Guy
%I A019040
%S A019040 1,20,281,3460,40161,453300,5048041,55853540,616079921,
%T A019040 6785596180,74686191801,821775473220,9040683799681,99453356876660,
%U A019040 1094016369479561,12034328357198500,132378357688767441
%N A019040 Expansion of 1/((1-4x)(1-5x)(1-11x)).
%K A019040 nonn
%O A019040 0,2
%A A019040 njas
%I A021204
%S A021204 1,20,281,3472,40509,459564,5139121,57034088,630398021,
%T A021204 6952517572,76586531385,843104877888,9278071860877,102082299710684,
%U A021204 1123046352296513,12354356208201112,135902996287980117
%N A021204 Expansion of 1/((1-x)(1-2x)(1-6x)(1-11x)).
%K A021204 nonn
%O A021204 0,2
%A A021204 njas
%I A017953
%S A017953 1,20,283,3518,41209,468608,5247271,58277666,644406997,
%T A017953 7108612676,78315612739,862197157094,9488521761265,104399859167624,
%U A017953 1148555174389087,12635047273900202,138991162189670413
%N A017953 Expansion of 1/((1-3x)(1-6x)(1-11x)).
%K A017953 nonn
%O A017953 0,2
%A A017953 njas
%I A016317
%S A016317 1,20,284,3520,40656,450240,4851904,51315200,535521536,
%T A016317 5534172160,56773377024,579187015680,5883496124416,59567968993280,
%U A016317 601543751942144,6062350015528960,60998800124215296,612990400993689600
%N A016317 Expansion of 1/((1-2x)(1-8x)(1-10x)).
%K A016317 nonn
%O A016317 0,2
%A A016317 njas
%I A021404
%S A021404 1,20,285,3640,44681,540540,6505045,78138080,937976961,
%T A021404 11257033060,135089723405,1621098253320,19453266126841,
%U A021404 233439544261580,2801275941271365,33615316957309360,403383826200494321
%N A021404 Expansion of 1/((1-x)(1-3x)(1-4x)(1-12x)).
%K A021404 nonn
%O A021404 0,2
%A A021404 njas
%I A046175
%S A046175 0,1,20,285,3976,55385,771420,10744501,149651600,2084377905,29031639076,
%T A046175 404358569165,5631988329240,78443478040201,1092576704233580,
%U A046175 15217630381229925,211954248632985376,2952141850480565345
%N A046175 Indices of triangular numbers which are also pentagonal.
%H A046175 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A046175 a(n) = 14*a(n-1) - a(n-2) + 6; g.f.: (1+5*x)/((1-x)*(1-14*x+x^2)) - Warut Roonguthai (warut@ksc9.th.com), Jan 05 2001
%Y A046175 Cf. A014979, A046174.
%K A046175 nonn,easy
%O A046175 0,3
%A A046175 Eric W. Weisstein (eric@weisstein.com)
%I A016314
%S A016314 1,20,287,3634,43329,500136,5666179,63480878,706360277,
%T A016314 7826457892,86486501751,954119775882,10514695335145,115797293297888,
%U A016314 1274719738572203,14028563710395046,154360726917116733
%N A016314 Expansion of 1/((1-2x)(1-7x)(1-11x)).
%K A016314 nonn
%O A016314 0,2
%A A016314 njas
%I A021164
%S A021164 1,20,287,3694,45609,553776,6677779,80295938,964364717,
%T A021164 11576444932,138937682871,1667353916982,20008755624625,
%U A021164 240107610616088,2881304043028763,34575712094589226,414908863026422133
%N A021164 Expansion of 1/((1-x)(1-2x)(1-5x)(1-12x)).
%K A021164 nonn
%O A021164 0,2
%A A021164 njas
%I A017918
%S A017918 1,20,289,3740,46321,563300,6797569,81762860,982121041,
%T A017918 11790305780,141507994849,1698217742780,20379222467761,
%U A017918 244553718979460,2934659879368129,35215994824839500,422592319303230481
%N A017918 Expansion of 1/((1-3x)(1-5x)(1-12x)).
%K A017918 nonn
%O A017918 0,2
%A A017918 njas
%I A016261
%S A016261 1,20,291,3730,44681,513240,5730271,62683550,675263061,
%T A016261 7188478660,75807419051,793377882570,8251512054241,85374719599280,
%U A016261 879483587504631,9026463398652790,92349281698986221,942254646401987100
%N A016261 Expansion of 1/((1-x)(1-9x)(1-10x)).
%K A016261 nonn
%O A016261 0,2
%A A016261 njas
%I A016309
%S A016309 1,20,292,3824,47824,585536,7096384,85576448,1029436672,
%T A016309 12368356352,148510974976,1782675894272,21395375902720,
%U A016309 256764101869568,3081286768672768,36976146501533696,443717989683232768
%N A016309 Expansion of 1/((1-2x)(1-6x)(1-12x)).
%K A016309 nonn
%O A016309 0,2
%A A016309 njas
%I A016259
%S A016259 1,20,293,3808,46569,549708,6346381,72206936,813450257,
%T A016259 9101344516,101341923189,1124578223184,12448896999865,137566159357244,
%U A016259 1518254091799517,16741005720753352,184472748615956193
%N A016259 Expansion of 1/((1-x)(1-8x)(1-11x)).
%K A016259 nonn
%O A016259 0,2
%A A016259 njas
%I A047795
%S A047795 1,1,1,20,295,871,196784,6287772,29169631,18200393741,1304183716981,
%T A047795 27109895360074,6212943553813622,1062831339757496245,85292203894284124100,
%U A047795 1487854700305245210924,1896933688279584387159631,377233175400513002923379973
%V A047795 1,1,-1,-20,295,871,-196784,6287772,29169631,-18200393741,1304183716981,
%W A047795 -27109895360074,-6212943553813622,1062831339757496245,-85292203894284124100,
%X A047795 -1487854700305245210924,1896933688279584387159631,-377233175400513002923379973
%N A047795 Sum_{k=0..n} C(n,k)*Stirling1(n,k)*Stirling2(n,k).
%K A047795 sign,done
%O A047795 0,4
%A A047795 njas
%I A001708 M5095 N2206
%S A001708 1,20,295,4025,54649,761166,11028590,167310220,2664929476,
%T A001708 44601786944,784146622896,14469012689040,279870212258064,5667093514231200
%N A001708 Generalized Stirling numbers.
%D A001708 Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%F A001708 E.g.f.: ( ln ( 1 - x ))^4 / 24 ( 1 - x )^2.
%K A001708 nonn
%O A001708 0,2
%A A001708 njas
%I A016255
%S A016255 1,20,297,3964,50369,624036,7625689,92469068,1116354417,
%T A016255 13443332212,161649541001,1942101373212,23321364646945,
%U A016255 279969412942148,3360424215557433,40330629408450796,484006324653740753
%N A016255 Expansion of 1/((1-x)(1-7x)(1-12x)).
%K A016255 nonn
%O A016255 0,2
%A A016255 njas
%I A053541
%S A053541 1,20,300,4000,50000,600000,7000000,80000000,900000000,10000000000,
%T A053541 110000000000,1200000000000,13000000000000,140000000000000,
%U A053541 1500000000000000,16000000000000000,170000000000000000
%N A053541 A second order recursive sequence.
%D A053541 A. H. Beiler, Recreations in the Theory of Numbers,Dover,N.Y.,1964,pps.194-196.
%H A053541 F. Ellermann, Illustration of binomial transforms
%F A053541 a(n)=n10^(n-1); a(n)=20a(n-1)-100a(n-2); a(0)=1; n>0.
%Y A053541 Cf. A001787, A053464 and A053469.
%K A053541 easy,nonn
%O A053541 0,2
%A A053541 Barry E. Williams, Jan 15 2000
%E A053541 More terms from Larry Reeves (larryr@acm.org), May 29 2001
%I A004345
%S A004345 1,20,300,4060,52360,658008,8145060,99884400,1217566350,
%T A004345 14783142660,179013799328,2163842859360,26123889412400,
%U A004345 315136287090800,3799541229226200,45795673964460816,551876736990451095
%N A004345 Binomial coefficient C(5n,n-3).
%D A004345 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.
%K A004345 nonn,easy
%O A004345 3,2
%A A004345 njas
%I A001755 M5096 N2207
%S A001755 1,20,300,4200,58800,846720,12700800,199584000,3293136000,
%T A001755 57081024000,1038874636800,19833061248000,396661224960000,
%U A001755 8299373322240000,181400588328960000,4135933413900288000,98228418580131840000
%N A001755 Lah numbers: n!C(n-1,3)/4!.
%D A001755 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
%D A001755 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
%p A001755 A001755:=n-> n!*binomial(n-1,3)/4!;
%Y A001755 Column 4 of A008297. Cf. A053495.
%K A001755 nonn,easy
%O A001755 4,2
%A A001755 njas
%E A001755 More terms from Barbara Haas Margolius (margolius@math.csuohio.edu) 2/12/01
%I A016190
%S A016190 1,20,301,4040,51001,620060,7352101,85656080,985263601,
%T A016190 11225320100,126965305501,1427999420120,15990423157801,
%U A016190 178436520564140,1985678518660501,22048354837360160,244384923399813601
%N A016190 Expansion of 1/((1-9x)(1-11x)).
%K A016190 nonn
%O A016190 0,2
%A A016190 njas
%I A016188
%S A016188 1,20,304,4160,54016,680960,8433664,103301120,1256390656,
%T A016188 15210905600,183604609024,2211845242880,26610862391296,
%U A016188 319880104509440,3842959300624384,46150695979581440,554089826731687936
%N A016188 Expansion of 1/((1-8x)(1-12x)).
%K A016188 nonn
%O A016188 0,2
%A A016188 njas
%I A006300 M5097
%S A006300 1,20,307,4280,56914,736568,9370183,117822512,1469283166,18210135416,
%T A006300 224636864830,2760899996816,33833099832484,413610917006000
%N A006300 Rooted maps with n edges on torus.
%D A006300 E. A. Bender, E. R. Canfield and R. W. Robinson, The enumeration of maps on the torus and the projective plane, Canad. Math. Bull., 31 (1988), 257-271.
%D A006300 T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.
%D A006300 T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
%Y A006300 Cf. A007137.
%K A006300 nonn,nice,more
%O A006300 2,2
%A A006300 njas
%E A006300 Bender et al. give 20 terms.
%I A034094
%S A034094 20,312,9744,29280,53352,1666224,5006880,106798080,980733600,133301760,
%T A034094 9099742080,22794600960,1556055895680,3577201689600,4464942451200,
%U A034094 380428773854896765462278360268800000
%N A034094 (-1)-sigma perfect numbers: (-1)sigma(a) = m*a for some integer m, where if a = Product p(i)^r(i) then (-1)sigma(a) = Product (-1+Sum p(i)^s(i), s(i)=1 to r(i)).
%C A034094 The indexes of some terms are 1, so these numbers are fixed points of (-1)sigma.
%e A034094 Factorizations 2^2*5, 2^3*3*13, 2^4*3*7*29, 2^5*3*5*61, 2^3*3^3*13*19, 2^4*3^3*7*19*29, 2^5*3^3*5*19*61, 2^10*3*5*17*409, 2^5*3^2*5^2*7*11*29*61, 2^9*3*5*17*1021, 2^7*3*5*11^2*13*23*131, 2^9*3^3*5*17*19*1021, 2^7*3^3*5*11^2*13*19*23*131, 2^10*3^2*5^2*7*11*17*29*409, 2^9*3^2*5^2*7*11*17*29*1021, 2^24*3^3*5^5*7^2*11*17*19*29*61*233*239*467*479*70051
%Y A034094 Cf. A034095.
%K A034094 nonn
%O A034094 0,1
%A A034094 Yasutoshi Kohmoto (kohmoto@z2.zzz.or.jp)
%I A011197
%S A011197 0,20,315,1820,6630,18480,43225,89320,168300,295260,489335,
%T A011197 774180,1178450,1736280,2487765,3479440,4764760,6404580,
%U A011197 8467635,11031020,14180670,18011840,22629585,28149240,34696900
%N A011197 n*(n+1)*(2*n+1)*(3*n+1)*(4*n+1)/6.
%K A011197 nonn
%O A011197 0,2
%A A011197 njas
%I A054621
%S A054621 0,1,20,315,2344,11165,39996,117655,299600,683289,1428580,2783891,
%T A054621 5118840,8964085,15059084,24408495,38347936,58619825,87460020,127695979,
%U A054621 182857160,257298381,356336860,486403655,655210224,871930825,1147401476
%N A054621 Sum_{d|7} phi(d)*n^(7/d)/7.
%K A054621 nonn
%O A054621 0,3
%A A054621 njas, Apr 16 2000
%I A024387
%S A024387 20,319,1850,6962,20344,50198,109666,218483,404885,707740,1178937,
%T A024387 1885998,2914945,4373393,6393898,9137529,12797693,17604194,23827533,
%U A024387 31783445,41837686,54411045,69984609,89105263,112391425,140539035
%N A024387 [ (4th elementary symmetric function of S(n))/(1st elementary symmetric function of S(n)) ], where S(n) = {first n+3 positive integers congruent to 1 mod 4}.
%K A024387 nonn
%O A024387 1,1
%A A024387 Clark Kimberling (ck6@cedar.evansville.edu)
%I A005748 M5098
%S A005748 1,20,348,6093,108182,1890123,31500927,490890277,7086257602,
%T A005748 94548676765,1167995082810,13406707973018,143598707530374,
%U A005748 1441525802509250,13619352767824724,121574625625030584
%N A005748 n-covers of a 7-set.
%D A005748 R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
%H A005748 Vladeta Jovovic, Binary matrices up to row and column permutations
%Y A005748 Cf. A005744-A005747, A005783-A005786, A055066.
%K A005748 nonn
%O A005748 1,2
%A A005748 njas, Simon Plouffe (plouffe@math.uqam.ca)
%E A005748 Corrected and extended by Vladeta Jovovic (vladeta@Eunet.yu), Jun 13 2000
%I A004292
%S A004292 1,20,360,6460,115920,2080100,37325880,669785740,12018817440,
%T A004292 215668928180,3870021889800,69444725088220,1246135029698160,
%U A004292 22360985809478660,401251609540917720,7200167985927040300
%N A004292 Expansion of (1+2*x+x^2)/(1-18*x+x^2).
%D A004292 J. M. Alonso, Growth functions of amalgams, in Alperin, ed., Arboreal Group Theory, Springer, pp. 1-34, esp. p. 32.
%D A004292 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.
%K A004292 nonn
%O A004292 0,2
%A A004292 njas
%I A053508
%S A053508 0,0,0,1,20,360,6860,143360,3306744,84000000,2338460520,70946979840,
%T A053508 2332989862060,82726831323136,3148511132812500,128071114403348480,
%U A053508 5546563698427324720,254873089955815096320,12387799656377835411984
%N A053508 binomial(n-1,3)*n^(n-4).
%D A053508 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Prop. 5.3.2.
%F A053508 E.g.f.: 1/4!*LambertW(-x)^4. - Vladeta Jovovic (vladeta@Eunet.yu), Apr 07 2001
%Y A053508 Cf. A000169, A053506-A053509.
%K A053508 nonn
%O A053508 1,5
%A A053508 njas, Jan 15 2000
%I A060918
%S A060918 1,20,360,6860,143570,3321864,84756000,2372001720,72384192540,
%T A060918 2394775746220,85443353291296,3271908306712500,133893717061821080,
%U A060918 5832748749666611920,269542701201588099840,13172225935626444660144
%N A060918 E.g.f.: exp((-1)^k/k*LambertW(-x)^k)/(k-1)!, k=4.
%F A060918 a(n)=(n-1)!/(k-1)!*Sum_{i=0..floor((n-k)/k)} 1/(i!*k^i)*n^(n-(i+1)*k)/(n-(i+1)*k)!, k=4.
%Y A060918 Cf. A057817, A060917.
%K A060918 easy,nonn
%O A060918 4,2
%A A060918 Vladeta Jovovic (vladeta@Eunet.yu), Apr 10 2001
%I A049683
%S A049683 0,1,20,361,6480,116281,2086580,37442161,671872320,12056259601,
%T A049683 216340800500,3882078149401,69661065888720
%N A049683 a(n)=(L(6n)-2)/16, where L=A000032 (the Lucas sequence).
%K A049683 nonn
%O A049683 0,3
%A A049683 Clark Kimberling, ck6@cedar.evansville.edu
%I A014901
%S A014901 1,20,363,6538,117689,2118408,38131351,686364326,12354557877,
%T A014901 222382041796,4002876752339,72051781542114,1296932067758065,
%U A014901 23344777219645184,420205989953613327,7563707819165039902
%N A014901 a(1)=1, a(n)=18*a(n-1)+n.
%K A014901 nonn
%O A014901 1,2
%A A014901 njas, Olivier Gerard (ogerard@ext.jussieu.fr)
%I A000564 M5099 N2208
%S A000564 20,371,2588,11097,35645,94457,218124,454220,872648,1571715,
%T A000564 2684936,4388567,6909867,10536089,15624200,22611330,32025950,44499779,
%U A000564 60780420,81744725,108412889,141963273,183747956,235309016,298395540
%N A000564 Discordant permutations.
%D A000564 J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
%F A000564 G.f.: -x^6(2x^7-4x^6+36x^5-29x^4-72x^3-411x^2-231x-20)/((1-x)^7).
%F A000564 a(n)=81/80n^6-405/16n^5+4113/16n^4-21267/16n^3+140357/40n^2-7587/2n, n>6.
%p A000564 rr:=n - >81/80*n^6 - 405/16*n^5 + 4113/16*n^4 - 21267/16*n^3 + 140357/40*n^2 - 7587/2*n; seq(rr(n),n=7..40);
%K A000564 nonn
%O A000564 3,1
%A A000564 njas
%E A000564 More terms, formulae and Maple code from Barbara Haas Margolius (margolius@math.csuohio.edu) 2/17/01
%I A019580
%S A019580 1,20,375,7560,167825,4110120
%N A019580 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives f(n,4)/n.
%Y A019580 Cf. A019576.
%K A019580 nonn
%O A019580 4,2
%A A019580 lcorbin@netcom.com (Lee Corbin)
%I A063815
%S A063815 1,20,380,7220,137180,2606420,49521980,940917620,17877434780,339671260820,
%T A063815 6453753955560,122621325155260,2329805177942740,44266298380775260,
%U A063815 841059669232130740,15980133715361099260,303622540590922574740
%N A063815 Growth series for fundamental group of orientable closed surface of genus 5.
%D A063815 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, Sigma(Gamma_5, S_5; z).
%F A063815 G.f.: (1+x)*(1-x^10)/(1-19*x+19*x^10-x^11).
%K A063815 nonn
%O A063815 0,2
%A A063815 njas, Aug 21 2001
%I A019319
%S A019319 1,20,400,5362,71852
%N A019319 Possible chess positions after n moves.
%D A019319 Bernd Schwarzkopf, ''Die ersten Z"uge'' (The First Moves), Problemkiste (No.92, April 1994, p.142-143).
%H A019319 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A019319 Cf. A006494.
%K A019319 fini,nonn,hard
%O A019319 0,2
%A A019319 Bernd Schwarzkopf (schwarzkopf@uni-duesseldorf.de)
%I A057745
%S A057745 1,20,400,5362,72078,822518,9417683
%N A057745 Number of distinct chess positions after n plies including differences due to availability of castling and en passant captures.
%Y A057745 Cf. A019319, A048987.
%K A057745 hard,nonn,nice
%O A057745 0,2
%A A057745 Steven J. Edwards (chessnotation@earthlink.net), Oct 29 2000 and Dec 16 2000.
%I A009964
%S A009964 1,20,400,8000,160000,3200000,64000000,1280000000,25600000000,
%T A009964 512000000000,10240000000000,204800000000000,4096000000000000,
%U A009964 81920000000000000,1638400000000000000,32768000000000000000
%N A009964 Powers of 20.
%F A009964 G.f.: 1/(1-20x), e.g.f.: exp(20x)
%K A009964 nonn,easy
%O A009964 0,2
%A A009964 njas
%I A007577
%S A007577 1,20,400,8902,197281,4865577,119057155,3194833218
%N A007577 Number of chess games with n moves (another version).
%Y A007577 Cf. A006494, A019319.
%K A007577 fini,nonn,hard
%O A007577 0,2
%A A007577 Johan Boye (jb@sectra.se)
%E A007577 I am not sure of the precise rules that were used to compute these numbers. A006494 and A048987 are the preferred versions of this sequence.
%I A048987
%S A048987 1,20,400,8902,197281,4865609,119060324,3195901860,84998978956,
%T A048987 2439530234167
%N A048987 Number of chess games that end in exactly n plies.
%C A048987 a(n) does not include games which end in less than n moves.
%Y A048987 Cf. A006494.
%K A048987 nonn
%O A048987 0,2
%A A048987 Steven J. Edwards (sje@mv.mv.com)
%I A006494
%S A006494 1,20,400,8902,197281,4865617,119060679,3195913043
%N A006494 Number of chess games that end in exactly n plies plus games that terminate (i.e. mate) in less than n plies.
%C A006494 Richard Stanley (rstan@math.mit.edu) comments that the description should say: Number of chess games that end in exactly n plies plus games that terminate (i.e. mate) in less than mLink to a section of The World of Mathematics.
%Y A006494 Cf. A019319, A048987.
%K A006494 nonn,hard,nice
%O A006494 0,2
%A A006494 Ken Thompson (ken@plan9.bell-labs.com)
%E A006494 This and A048987 are the preferred versions of this sequence.
%I A007545 M5100
%S A007545 1,20,400,8902,197742,4897256,120921506,3284294545,88867026005
%N A007545 Number of chess games with n moves (another version).
%H A007545 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A007545 Cf. A006494, A019319.
%K A007545 fini,nonn,hard
%O A007545 0,2
%A A007545 Ken Thompson (ken@plan9.bell-labs.com)
%E A007545 I am not sure of the precise rules that were used to compute these numbers. A006494 and A048987 are the preferred versions of this sequence.
%I A055476
%S A055476 1,20,400,13000,310000,11200000,224000000,10030000000,201100000000,
%T A055476 4022000000000,130440000000000,3114300000000000,112341000000000000,
%U A055476 2302320000000000000,101101400000000000000,2022033000000000000000
%N A055476 Powers of ten written in base 5.
%Y A055476 Cf. A000468, A011557.
%K A055476 base,easy,nonn
%O A055476 0,2
%A A055476 Henry Bottomley (se16@btinternet.com), Jun 27 2000
%E A055476 More terms from James A. Sellers (sellersj@math.psu.edu), Jul 04 2000
%I A041181
%S A041181 1,20,401,8040,161201,3232060,64802401,1299280080,26050404001,
%T A041181 522307360100,10472197606001,209966259480120,4209797387208401,
%U A041181 84405914003648140,1692328077460171201,33930967463207072160
%N A041181 Denominators of continued fraction convergents to sqrt(101).
%Y A041181 Cf. A041180.
%K A041181 nonn,cofr,easy
%O A041181 0,2
%A A041181 njas
%I A041762
%S A041762 20,401,16060,321601,12880100,257923601,10329824140,206854406401,
%T A041762 8284506080180,165896976010001,6644163546480220,133049167905614401,
%U A041762 5328610879771056260,106705266763326739601,4273539281412840640300
%N A041762 Numerators of continued fraction convergents to sqrt(402).
%Y A041762 Cf. A041763.
%K A041762 nonn,cofr,easy
%O A041762 0,1
%A A041762 njas
%I A049382
%S A049382 1,20,450,10500,249375,5985000,144637500,3512625000,85620234375,
%T A049382 2092939062500,51277007031250,1258617445312500,30941012197265625,
%U A049382 761624915625000000,18768613992187500000,462959145140625000000
%N A049382 Expansion of (1-x)^(-4/5).
%F A049382 g.f. (1-25*x)^(-4/5).
%F A049382 a(n) = 5^n/n! * product[ k=0..n-1 ] (5*k+4).
%F A049382 a(n) ~ Gamma(4/5)^-1*n^(-1/5)*5^(2*n)*{1 - 2/25*n^-1 + ...}. - Joe Keane (jgk@jgk.org), Nov 24 2001
%e A049382 (1-x)^(-4/5) = 1 + 4/5*x + 18/25*x^2 + 84/125*x^3 + ...
%Y A049382 Cf. A008546, A049393, A049397, A049381.
%K A049382 nonn,easy
%O A049382 0,2
%A A049382 Joe Keane (jgk@jgk.org)
%I A065412
%S A065412 0,0,0,0,1,20,480,10560
%N A065412 (A005867 - A000165)/96.
%C A065412 A005867 can be generated recursively using A006093, and A000165 can be generated recursively using the even numbers. Since all primes after 2 are odd, (A005867 - A000165)/96 is an integer.
%e A065412 A005867(n+1) begins 1 2 8 48 480 5760 92160 ... and A000165(n) begins 1 2 8 48 384 3840 46080 ... so their difference begins 0 0 0 0 96 1920 46080 ...
%Y A065412 Cf. A000165, A005867, A006093.
%K A065412 easy,more,nonn
%O A065412 0,6
%A A065412 arnold1940 (Arnold1940@aol.com), Nov 23 2001
%I A000827 M5101 N2209
%S A000827 20,484,497760,1957701217328
%N A000827 Switching networks.
%D A000827 M. A. Harrison, On the number of classes of switching networks, J. Franklin Instit., 276 (1963), 313-327.
%H A000827 Index entries for sequences related to switching networks
%K A000827 nonn
%O A000827 1,1
%A A000827 njas
%I A008270
%S A008270 1,20,507,19552,1113485,88725876,9452410135,1299140690912,
%T A008270 223938108997497,47323772172058420,12033854264863090451,
%U A008270 3625294706255832787200,1276951433895343148472517
%N A008270 Total length of strings on n symbols in Stockhausen problem.
%D A008270 R. C. Read, Combinatorial problems in theory of music, Discrete Math. 167 (1997), 543-551.
%D A008270 Read, Ronald C. and Yen, Lily, A note on the Stockhausen problem, J. Comb. Theory, Ser. A 76, No.1, 1-10.
%K A008270 nonn
%O A008270 1,2
%A A008270 Lily Yen [ lyen@jeeves.uwaterloo.ca ]
%I A047682
%S A047682 20,544,66560,17895424,8329625600,5937093935104,6004799480791040,
%T A047682 8176700159200067584,14421891569272362106880,31983597922505761818148864,
%U A047682 87107695717210805652024197120,285816431841945942589104606674944
%V A047682 20,-544,66560,-17895424,8329625600,-5937093935104,6004799480791040,
%W A047682 -8176700159200067584,14421891569272362106880,-31983597922505761818148864,
%X A047682 87107695717210805652024197120,-285816431841945942589104606674944
%N A047682 4^(2*n)*(4^(2*n)-1)*bernoulli(2*n)/(2*n).
%D A047682 J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 283.
%H A047682 Index entries for sequences related to Bernoulli numbers.
%K A047682 sign,done
%O A047682 0,1
%A A047682 njas
%I A012568
%S A012568 1,0,20,560,7920,1034880,172569280,12339608320,1902800305920,
%T A012568 1037015219466240,208832669083243520,24402030281141555200,
%U A012568 43057075784745711636480,18939654540025835056496640
%V A012568 1,0,-20,-560,-7920,1034880,172569280,12339608320,-1902800305920,
%W A012568 -1037015219466240,-208832669083243520,24402030281141555200,
%X A012568 43057075784745711636480,18939654540025835056496640
%N A012568 arctanh(sinh(x)*cos(x))=x-20/5!*x^5-560/7!*x^7-7920/9!*x^9...
%K A012568 sign,done
%O A012568 0,3
%A A012568 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A027407
%S A027407 1,20,590,21680,935770,45961096,2515881220,151228367840,9871829559980,
%T A027407 693789631115120,52132594993215496,4164523150393491520,
%U A027407 351983138874505854040,31348448683850114451680
%N A027407 Labeled servers of dimension 20.
%D A027407 R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
%F A027407 E.g.f.: Exp[ Sum[ ((1+x)^i-1)/i,{i, 1, 20} ] ]
%K A027407 nonn
%O A027407 0,2
%A A027407 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A035279
%S A035279 1,20,600,24000,1200000,72000000,5040000000,403200000000,
%T A035279 36288000000000,3628800000000000,399168000000000000,
%U A035279 47900160000000000000,6227020800000000000000,871782912000000000000000
%N A035279 One tenth of deca-factorial numbers.
%C A035279 E.g.f. is G.f. of A011557(n-1) (powers of ten).
%H A035279 Index entries for sequences related to factorial numbers
%F A035279 10*a(n) = (10*n)(!^10):= product(10*j,j=1..n)= 10^n*n!; E.g.f. (-1+(1-10*x)^(-1))/10.
%Y A035279 Cf. A011557, A045757.
%K A035279 easy,nonn
%O A035279 1,2
%A A035279 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A015268
%S A015268 1,20,610,15860,433771,11662040,315323620,8509702520,229798289941,
%T A015268 6204226946060,167517069529030,4522934399547980,122119467087816511,
%U A015268 3297223466672052080,89025052902439936840,2403676254645238280240
%V A015268 1,-20,610,-15860,433771,-11662040,315323620,-8509702520,229798289941,
%W A015268 -6204226946060,167517069529030,-4522934399547980,122119467087816511,
%X A015268 -3297223466672052080,89025052902439936840,-2403676254645238280240
%N A015268 Gaussian binomial coefficient [ n,3 ] for q=-3.
%D A015268 J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
%D A015268 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p, 99.
%D A015268 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%K A015268 sign,done,easy,huge
%O A015268 3,2
%A A015268 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A059420
%S A059420 1,20,616,28160,1805056,154918400,17171485696,2389096202240,
%T A059420 407776690241536,83793407533383680,20407552701432856576,
%U A059420 5813146553630295326720,1914890082050142493474816
%N A059420 A diagonal of A059419.
%D A059420 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.
%F A059420 a(n)= A059419(2*n+1,3). From the expansion of tan(x)^3/6.
%Y A059420 Cf. A059419.
%K A059420 nonn,easy,huge
%O A059420 1,2
%A A059420 njas, Jan 30 2001
%E A059420 More terms from Larry Reeves (larryr@acm.org), Feb 08 2001. Corrected and extended by Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Feb 09 2001.
%I A006410 M5102
%S A006410 20,651,8344,64667,361884,1607125,5997992
%N A006410 Rooted nonseparable maps on the torus.
%D A006410 Walsh, T. R. S.; Lehman, A. B.; Counting rooted maps by genus. III: Nonseparable maps. J. Combinatorial Theory Ser. B 18 (1975), 222-259.
%K A006410 nonn
%O A006410 2,1
%A A006410 njas
%I A034404
%S A034404 20,680,29260,34220,70300,221815,227920,287980,467180,908600,
%T A034404 2481115,4278680,12259940,13813570,15493204,17861900,19970444,
%U A034404 24672560,25665020,27880600,29742164,34055980,44722580
%N A034404 Values of C(n,3) which can be written as C(x,3)+C(y,3).
%C A034404 Bombieri's Napkin Problem: Bombieri said that "the equation C(x,n)+C(y,n)=C(z,n) has no trivial solutions for n >= 3" (the joke being that he said "trivial" rather than "nontrivial"!).
%D A034404 Van der Poorten, Notes on Fermat's Last Theorem, Wiley, p. 122.
%e A034404 C(10,3)+C(16,3)=C(17,3)=680.
%Y A034404 Cf. A010330.
%K A034404 nonn,nice
%O A034404 0,1
%A A034404 njas
%I A012802
%S A012802 1,0,20,784,26256,4466176,435729216,104111460608,23261262303488,
%T A012802 7794968497086464,2988775301503636480,1434421552426973564928,
%U A012802 811112609409475303936000,540598484103424258178285568
%V A012802 1,0,20,-784,26256,-4466176,435729216,-104111460608,23261262303488,
%W A012802 -7794968497086464,2988775301503636480,-1434421552426973564928,
%X A012802 811112609409475303936000,-540598484103424258178285568
%N A012802 sin(sec(x)*arctan(x))=x+20/5!*x^5-784/7!*x^7+26256/9!*x^9...
%K A012802 sign,done
%O A012802 0,3
%A A012802 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A049214
%S A049214 1,20,784,52352,5360256,782525952,154594381824,39746508226560,
%T A049214 12902483299368960,5162443736924160000,2496471943395999744000,
%U A049214 1435556519572510605312000,968234590214616380866560000
%N A049214 Scaled coefficients of (arctanh x)^3.
%F A049214 e.g.f. (arctanh x)^3 or (1-x^2)^-1 * (arctanh x)^2
%F A049214 a(n) is coefficient of x^(2*n+3) in (arctanh x)^3, multiplied by (2*n+3)!/6
%e A049214 (arctanh x)^3 = x^3 + x^5 + 14/15*x^7 + 818/945*x^9 + ...
%K A049214 nonn
%O A049214 0,2
%A A049214 Joe Keane (jgk@jgk.org)
%I A002455 M5103 N2210
%S A002455 0,1,20,784,52480,5395456,791691264,157294854144,40683662475264,
%T A002455 13288048674471936,5349739088314368000,2603081566154391552000,
%U A002455 1506057980251484454912000,1021944601582419125993472000
%N A002455 Central factorial numbers.
%D A002455 T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7.
%D A002455 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. 110.
%D A002455 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A002455 Ramanujan's Notebooks, Part I, by B. Berndt, page 263.
%H A002455 Index entries for sequences related to factorial numbers
%F A002455 E.g.f.: (arcsin x)^4; that is, a_k is the coefficient of x^(2*k+4) in (arcsin x)^4 multiplied by (2*k+4)! and divided by 4! Also a(n) = 2^(2*n-2)*(n!)^2 * sum[ k=1..n ] k^(-2) - from Joe Keane (jgk@jgk.org)
%e A002455 (arcsin x)^4 = x^4 + 2/3*x^6 + 7/15*x^8 + 328/945*x^10 + ...
%Y A002455 Cf. AA001819, A001824, A001825, A049033.
%K A002455 nonn,easy,nice
%O A002455 0,3
%A A002455 njas
%E A002455 More terms from Joe Keane (jgk@jgk.org)
%I A041763
%S A041763 1,20,801,16040,642401,12864060,515204801,10316960080,413193608001,
%T A041763 8274189120100,331380758412001,6635889357360120,265766955052816801,
%U A041763 5321974990413696140,213144766571600662401,4268217306422426944160
%N A041763 Denominators of continued fraction convergents to sqrt(402).
%Y A041763 Cf. A041762.
%K A041763 nonn,cofr,easy
%O A041763 0,2
%A A041763 njas
%I A041760
%S A041760 20,801,32060,1283201,51360100,2055687201,82278848140,3293209612801,
%T A041760 131810663360180,5275719744020001,211160600424160220,8451699736710428801,
%U A041760 338279150068841312260,13539617702490362919201,541922987249683358080300
%N A041760 Numerators of continued fraction convergents to sqrt(401).
%Y A041760 Cf. A041761.
%K A041760 nonn,cofr,easy
%O A041760 0,1
%A A041760 njas
%I A006424 M5104
%S A006424 20,831,12656,109075,648792,2978245,11293436,36973989
%N A006424 Rooted toroidal maps.
%D A006424 Walsh, T. R. S.; Lehman, A. B.; Counting rooted maps by genus. III: Nonseparable maps. J. Combinatorial Theory Ser. B 18 (1975), 222-259.
%K A006424 nonn
%O A006424 1,1
%A A006424 njas
%I A066802
%S A066802 20,924,48620,2704156,155117520,9075135300,538257874440,32247603683100,
%T A066802 1946939425648112,118264581564861424,7219428434016265740,
%U A066802 442512540276836779204,27217014869199032015600
%N A066802 Binomial(6*n,3*n).
%K A066802 nonn
%O A066802 1,1
%A A066802 Benoit Cloitre (abcloitre@wanadoo.fr), Jan 18 2002
%I A066798
%S A066798 20,944,49564,2753720,157871240,9233006540,547490880980,32795094564080,
%T A066798 1979734520212192,120244316085073616,7339672750101339356,
%U A066798 449852213026938118560,27666867082225970134160
%N A066798 Sum(i=1,n,binomial(6*i,3*i)).
%K A066798 nonn
%O A066798 1,1
%A A066798 Benoit Cloitre (abcloitre@wanadoo.fr), Jan 18 2002
%I A006427 M5105
%S A006427 20,1071,26320,431739,5494896,58677420,550712668,4681144391
%N A006427 Rooted toroidal maps.
%D A006427 Walsh, T. R. S.; Lehman, A. B.; Counting rooted maps by genus. III: Nonseparable maps. J. Combinatorial Theory Ser. B 18 (1975), 222-259.
%K A006427 nonn
%O A006427 1,1
%A A006427 njas
%I A002305 M5106 N2211
%S A002305 1,20,1120,3200,3942400,66560000,10035200000
%N A002305 Denominators of coefficients in asymptotic expansion of (2/pi)*Integral_{0..inf} (sin x / x)^n dx.
%D A002305 R. G. Medhurst and J. H. Roberts, Evaluation of the integral I_n(b) = of (2/pi)*Integral_{0..inf} (sin x / x)^n cos (bx) dx, Math. Comp., 19 (1965), 113-117.
%Y A002305 Cf. A002304, A002297, A002298.
%K A002305 nonn,frac,easy
%O A002305 0,2
%A A002305 njas
%I A042745
%S A042745 1,20,1201,24040,1443601,28896060,1735207201,34733040080,
%T A042745 2085717612001,41749085280100,2507030834418001,50182365773640120,
%U A042745 3013448977252825201,60319161910830144140,3622163163627061473601
%N A042745 Denominators of continued fraction convergents to sqrt(903).
%Y A042745 Cf. A042744.
%K A042745 nonn,cofr,easy
%O A042745 0,2
%A A042745 njas
%I A036067
%S A036067 0,0,20,1220,321120,13423120,1433424120,4443424120,8433423120,
%T A036067 187443422120,28175443523120,2827255433624120,282716454443823120,
%U A036067 382726356443923120,19382736256463823120,29382756253483823120
%N A036067 A summarize Fibonacci sequence: summarize the previous two terms!.
%C A036067 From the 54th term the sequence goes into a cycle of 117 terms.
%Y A036067 Cf. A036059.
%K A036067 base,easy,nice,nonn
%O A036067 0,3
%A A036067 Floor van Lamoen (f.v.lamoen@wxs.nl)
%I A001451
%S A001451 1,20,1260,100100,8817900,823727520,79919739900,7962100660800,
%T A001451 808906548235500,83426304143982800,8707404737345073760,
%U A001451 917663774856743842200,97491279924241456098300
%N A001451 (5*n)!/((3*n)!*n!*n!).
%p A001451 f:=n->(5*n)!/((3*n)!*n!*n!);
%K A001451 nonn
%O A001451 0,2
%A A001451 njas
%I A005982 M5107
%S A005982 1,20,1301,202840,61889101,32676403052,27418828825961,34361404413755056,
%T A005982 61335081309931829401,150221740688275657957940,489799709605132718770274141
%N A005982 3 up, 3 down, 3 up, ... permutations of length 2n+1.
%D A005982 P. R. Stein, personal communication.
%K A005982 nonn
%O A005982 1,2
%A A005982 njas
%I A014606
%S A014606 1,1,20,1680,369600,168168000,137225088000,182509367040000,
%T A014606 369398958888960000,1080491954750208000000,4386797336285844480000000,
%U A014606 23934366266775567482880000000,170891375144777551827763200000000
%N A014606 (3n)!/(6^n).
%K A014606 nonn
%O A014606 0,3
%A A014606 BjornE (mdeans@algonet.se)
%I A064878
%S A064878 1,1,20,2300,464000,116250000,32580600000,9779307000000,
%T A064878 3074524280000000,999451946900000000,333207298730000000000,
%U A064878 113305219025110000000000,39145823948711200000000000
%N A064878 Generalized Catalan numbers C(10,10;n).
%C A064878 See triangle A064879 with columns m built from C(m,m;n), m >= 0, also for Derrida et al. and Liggett refs.
%F A064878 a(n)= ((10^(2*(n-1)))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m,n-2-m)*((1/10)^(m+1)),m=0..n-2), n >= 2, a(0):=1=:a(1).
%F A064878 G.f.:(1-19*x*c(100*x))/(1-10*x*c(100*x))^2 = c(100*x)*(19+81*c(100*x))/(1+9*c(100*x))^2 = (19*c(100*x)*(10*x)^2+9*(9+29*x))/(9+10*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
%Y A064878 A064347.
%K A064878 nonn,easy
%O A064878 0,3
%A A064878 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Oct 12 2001
%I A028458
%S A028458 20,3108,297296,31937617,3350337360,353544256465,37264079953664,
%T A028458 3928877032323636,414211112701109036,43669866502128209569,
%U A028458 4604058193083952217472,485400369515158307543761
%N A028458 Number of perfect matchings in graph P_{2} X C_{6} X P_{n}.
%D A028458 P.H. Lundow, "Computation of matching polynomials and the number of 1-factors in polygraphs", Research report, No 12, 1996, Department of Math., Umea University, Sweden.
%H A028458 P.-H. Lundow, Enumeration of matchings in polygraphs, 1998.
%K A028458 nonn
%O A028458 1,1
%A A028458 Per-Hakan Lundow (per-hakan.lundow@math.umu.se)
%I A013725
%S A013725 20,8000,3200000,1280000000,512000000000,204800000000000,
%T A013725 81920000000000000,32768000000000000000,13107200000000000000000,
%U A013725 5242880000000000000000000,2097152000000000000000000000
%N A013725 20^(2n+1).
%K A013725 nonn,easy
%O A013725 0,1
%A A013725 njas
%I A045811
%S A045811 20,10022,10202,12002,12020,12200,20012,20102,20120,20201,20210,20220,
%T A045811 21002,21200,22002,22010,22020,22100,22200,1210020,1210022,1212200,
%U A045811 1220100,10222011,12200220,12202020,12210210,20010121,20010212
%N A045811 In the list of divisors of n (in base 3), each digit 0-2 appears equally often.
%C A045811 E.g. divisors of 12200 are (1, 10, 100, 122, 1220, 12200); the numbers of digits (0-2) are [ 0(6),1(6),2(6) ]
%H A045811 N. Nomoto, In the list of divisors of n,...
%Y A045811 Cf. A038564, A038565, A045812.
%K A045811 easy,nonn
%O A045811 0,1
%A A045811 Naohiro Nomoto (6284968128@geocities.co.jp)
%I A028667
%S A028667 1,20,12000,186000000,72540000000000,708171750000000000000,
%T A028667 172882428468750000000000000000,
%U A028667 1055177097007236328125000000000000000000
%N A028667 Galois numbers for p=5; order of group AGL(n,5).
%F A028667 a(n) = p^n Product (p^n - p^k) for k=0 to n-1
%t A028667 FoldList[ #1*5^#2 (5^#2-1)&,1,Range[ 20 ] ]
%K A028667 nonn
%O A028667 0,2
%A A028667 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A060618
%S A060618 0,1,20,22528
%N A060618 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=8 and D varies.
%D A060618 A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
%D A060618 N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octogonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
%D A060618 Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.
%H A060618 M. Latapy, Tilings of Zonotopes
%e A060618 For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 1.
%Y A060618 Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.
%K A060618 nonn
%O A060618 8,3
%A A060618 Matthieu Latapy (latapy@liafa.jussieu.fr), Apr 13 2001
%I A064487
%S A064487 20,29120,32537600,34093383680,35115786567680,36011213418659840,
%T A064487 36888985097480437760,37777778976635853209600,38685331082014736871587840,
%U A064487 39614005699412557795646504960,40564799864499450381466515537920
%N A064487 Order of twisted Suzuki group Sz(2^(2*n + 1)), also known as the group 2B2(2^(2*n + 1)).
%D A064487 R. W. Carter, Simple Groups of Lie Type, Wiley 1972, Chap. 14.
%D A064487 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985, p. xvi.
%F A064487 q^4*(q^2-1)*(q^4+1), where q^2 = 2^(2*n + 1).
%o A064487 (GAP) g:=Sz(32); s:=Size(g);
%Y A064487 Cf. A037250, A064583.
%K A064487 nonn
%O A064487 0,1
%A A064487 Dan Fux (danfux@my-deja.com), Oct 15 2001
%I A013766
%S A013766 20,160000,1280000000,10240000000000,81920000000000000,
%T A013766 655360000000000000000,5242880000000000000000000,41943040000000000000000000000,
%U A013766 335544320000000000000000000000000,2684354560000000000000000000000000000
%N A013766 20^(3n+1).
%K A013766 nonn
%O A013766 0,1
%A A013766 njas
%I A056104
%S A056104 20,1484110,15936368770,59961701958816,121740972715475096,
%T A056104 167109117756164222210,176340421320592288902670,
%U A056104 154794453668193645059165412,118987888829384136293188343172
%N A056104 9-antichain covers of a labeled n-set.
%D A056104 V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
%D A056104 V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
%H A056104 K. S. Brown, Dedekind's problem
%H A056104 E. W. Weisstein, Antichain covers"
%Y A056104 Cf. A056046-A056049, A056052, A051112-A051118.
%K A056104 huge,nonn
%O A056104 6,1
%A A056104 Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic (vladeta@Eunet.yu), Jul 29 2000
%I A051117
%S A051117 0,0,0,0,0,20,1484230,15946757960,60089234465176,122281201867047920,
%T A051117 168329227672583040430,178185327268349957044060,
%U A051117 156921594738520322214197672,121014019160263331691800711500
%N A051117 Monotone Boolean functions of n variables with 9 mincuts.
%D A051117 J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
%D A051117 V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
%D A051117 V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, Belgrade, 1999, in preparation.
%H A051117 K. S. Brown, Dedekind's Problem
%H A051117 Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
%H A051117 Index entries for sequences related to Boolean functions
%Y A051117 Cf. A016269, A047707, A051112-A051118.
%K A051117 nonn,easy,huge
%O A051117 0,6
%A A051117 Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic (vladeta@Eunet.yu)
%I A013812
%S A013812 20,3200000,512000000000,81920000000000000,13107200000000000000000,
%T A013812 2097152000000000000000000000,335544320000000000000000000000000,
%U A013812 53687091200000000000000000000000000000,8589934592000000000000000000000000000000000
%N A013812 20^(4n+1).
%K A013812 nonn
%O A013812 0,1
%A A013812 njas
%I A013894
%S A013894 20,64000000,204800000000000,655360000000000000000,2097152000000000000000000000,
%T A013894 6710886400000000000000000000000000,21474836480000000000000000000000000000000,
%U A013894 68719476736000000000000000000000000000000000000,219902325555200000000000000000000000000000000000000000
%N A013894 20^(5n+1).
%K A013894 nonn
%O A013894 0,1
%A A013894 njas
%I A048948
%S A048948 1,20,1292054400,10927088640,25841088000,218541772800,346787400960,
%T A048948 4084563974400,4422275520000,6935748019200,12161257680000,
%U A048948 31925327616000,133091939635200,233195801904000,2661838792704000
%N A048948 Values of y in solutions to sigma(x^3)=y^2, where x are given by A008849
%H A048948 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A048948 Cf. A008849.
%K A048948 nonn
%O A048948 1,2
%A A048948 Eric W. Weisstein (eric@weisstein.com)
%I A013547
%S A013547 1,0,0,21,0,3234,277,5856,303280,2437972574,588899927,13407976985576,
%T A013547 2342711674594,183937462367205,18926632135623132
%N A013547 sec(cosec(x)-cotanh(x))=1+21/6!*x^6+3234/10!*x^10+277/12!*x^12...
%K A013547 nonn
%O A013547 0,4
%A A013547 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A040445
%S A040445 21,1,1,1,1,3,3,21,3,3,1,1,1,1,42,1,1,1,1,3,3,21,3,3,1,1,1,1,42,
%T A040445 1,1,1,1,3,3,21,3,3,1,1,1,1,42,1,1,1,1,3,3,21,3,3,1,1,1,1,42,1,
%U A040445 1,1,1,3,3,21,3,3,1,1,1,1,42,1,1,1,1,3,3,21,3,3,1,1,1,1,42,1,1
%N A040445 Continued fraction for sqrt(467).
%H A040445 Index entries for continued fractions for constants
%p A040445 with(numtheory): Digits:=300: convert(evalf(sqrt(467)),confrac);
%K A040445 nonn,cofr,easy
%O A040445 0,1
%A A040445 njas
%I A040446
%S A040446 21,1,1,1,2,1,1,1,42,1,1,1,2,1,1,1,42,1,1,1,2,1,1,1,42,1,1,1,2,
%T A040446 1,1,1,42,1,1,1,2,1,1,1,42,1,1,1,2,1,1,1,42,1,1,1,2,1,1,1,42,1,
%U A040446 1,1,2,1,1,1,42,1,1,1,2,1,1,1,42,1,1,1,2,1,1,1,42,1,1,1,2,1,1,1
%N A040446 Continued fraction for sqrt(468).
%H A040446 Index entries for continued fractions for constants
%p A040446 with(numtheory): Digits:=300: convert(evalf(sqrt(468)),confrac);
%K A040446 nonn,cofr,easy
%O A040446 0,1
%A A040446 njas
%I A040447
%S A040447 21,1,1,1,10,6,10,1,1,1,42,1,1,1,10,6,10,1,1,1,42,1,1,1,10,6,10,
%T A040447 1,1,1,42,1,1,1,10,6,10,1,1,1,42,1,1,1,10,6,10,1,1,1,42,1,1,1,10,
%U A040447 6,10,1,1,1,42,1,1,1,10,6,10,1,1,1,42,1,1,1,10,6,10,1,1,1,42,1
%N A040447 Continued fraction for sqrt(469).
%H A040447 Index entries for continued fractions for constants
%p A040447 with(numtheory): Digits:=300: convert(evalf(sqrt(469)),confrac);
%K A040447 nonn,cofr,easy
%O A040447 0,1
%A A040447 njas
%I A040444
%S A040444 21,1,1,2,2,1,2,5,1,3,1,20,1,3,1,5,2,1,2,2,1,1,42,1,1,2,2,1,2,5,
%T A040444 1,3,1,20,1,3,1,5,2,1,2,2,1,1,42,1,1,2,2,1,2,5,1,3,1,20,1,3,1,5,
%U A040444 2,1,2,2,1,1,42,1,1,2,2,1,2,5,1,3,1,20,1,3,1,5,2,1,2,2,1,1,42,1
%N A040444 Continued fraction for sqrt(466).
%H A040444 Index entries for continued fractions for constants
%p A040444 with(numtheory): Digits:=300: convert(evalf(sqrt(466)),confrac);
%K A040444 nonn,cofr,easy
%O A040444 0,1
%A A040444 njas
%I A040443
%S A040443 21,1,1,3,2,2,2,3,1,1,42,1,1,3,2,2,2,3,1,1,42,1,1,3,2,2,2,3,1,1,
%T A040443 42,1,1,3,2,2,2,3,1,1,42,1,1,3,2,2,2,3,1,1,42,1,1,3,2,2,2,3,1,1,
%U A040443 42,1,1,3,2,2,2,3,1,1,42,1,1,3,2,2,2,3,1,1,42,1,1,3,2,2,2,3,1,1
%N A040443 Continued fraction for sqrt(465).
%H A040443 Index entries for continued fractions for constants
%p A040443 with(numtheory): Digits:=300: convert(evalf(sqrt(465)),confrac);
%K A040443 nonn,cofr,easy
%O A040443 0,1
%A A040443 njas
%I A040442
%S A040442 21,1,1,5,1,1,1,5,1,1,42,1,1,5,1,1,1,5,1,1,42,1,1,5,1,1,1,5,1,1,
%T A040442 42,1,1,5,1,1,1,5,1,1,42,1,1,5,1,1,1,5,1,1,42,1,1,5,1,1,1,5,1,1,
%U A040442 42,1,1,5,1,1,1,5,1,1,42,1,1,5,1,1,1,5,1,1,42,1,1,5,1,1,1,5,1,1
%N A040442 Continued fraction for sqrt(464).
%H A040442 Index entries for continued fractions for constants
%p A040442 with(numtheory): Digits:=300: convert(evalf(sqrt(464)),confrac);
%K A040442 nonn,cofr,easy
%O A040442 0,1
%A A040442 njas
%I A040441
%S A040441 21,1,1,13,1,5,4,1,1,1,1,2,2,6,1,3,21,3,1,6,2,2,1,1,1,1,4,5,1,13,
%T A040441 1,1,42,1,1,13,1,5,4,1,1,1,1,2,2,6,1,3,21,3,1,6,2,2,1,1,1,1,4,5,
%U A040441 1,13,1,1,42,1,1,13,1,5,4,1,1,1,1,2,2,6,1,3,21,3,1,6,2,2,1,1,1
%N A040441 Continued fraction for sqrt(463).
%H A040441 Index entries for continued fractions for constants
%p A040441 with(numtheory): Digits:=300: convert(evalf(sqrt(463)),confrac);
%K A040441 nonn,cofr,easy
%O A040441 0,1
%A A040441 njas
%I A022184
%S A022184 1,1,1,1,21,1,1,421,421,1,1,8421,168821,8421,1,1,168421,
%T A022184 67536821,67536821,168421,1,1,3368421,27014896821,540362104821,
%U A022184 27014896821,3368421,1,1,67368421,10805962096821,4322923853464821
%N A022184 Triangle of Gaussian binomial coefficients [ n,k ] for q = 20.
%D A022184 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%K A022184 nonn,tabl
%O A022184 0,5
%A A022184 njas
%I A015147
%S A015147 1,1,1,1,21,1,1,463,463,1,1,10185,224555,10185,1,1,224071,108674435,
%T A015147 108674435,224071,1,1,4929561,52598650611,1157056709445,52598650611,
%U A015147 4929561,1,1,108450343,25457741966163,12320392440820971
%V A015147 1,1,1,1,-21,1,1,463,463,1,1,-10185,224555,-10185,1,1,224071,108674435,
%W A015147 108674435,224071,1,1,-4929561,52598650611,-1157056709445,52598650611,
%X A015147 -4929561,1,1,108450343,25457741966163,12320392440820971
%N A015147 Triangle of (Gaussian) q-binomial coefficients for q=-22.
%K A015147 sign,done,tabl,easy
%O A015147 0,5
%A A015147 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A040450
%S A040450 21,1,2,1,1,1,4,5,4,1,1,1,2,1,42,1,2,1,1,1,4,5,4,1,1,1,2,1,42,1,
%T A040450 2,1,1,1,4,5,4,1,1,1,2,1,42,1,2,1,1,1,4,5,4,1,1,1,2,1,42,1,2,1,
%U A040450 1,1,4,5,4,1,1,1,2,1,42,1,2,1,1,1,4,5,4,1,1,1,2,1,42,1,2,1,1,1
%N A040450 Continued fraction for sqrt(472).
%H A040450 Index entries for continued fractions for constants
%p A040450 with(numtheory): Digits:=300: convert(evalf(sqrt(472)),confrac);
%K A040450 nonn,cofr,easy
%O A040450 0,1
%A A040450 njas
%I A040451
%S A040451 21,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,
%T A040451 42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,
%U A040451 42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1,42,1,2,1
%N A040451 Continued fraction for sqrt(473).
%H A040451 Index entries for continued fractions for constants
%p A040451 with(numtheory): Digits:=300: convert(evalf(sqrt(473)),confrac);
%K A040451 nonn,cofr,easy
%O A040451 0,1
%A A040451 njas
%I A040449
%S A040449 21,1,2,2,1,3,4,14,4,3,1,2,2,1,42,1,2,2,1,3,4,14,4,3,1,2,2,1,42,
%T A040449 1,2,2,1,3,4,14,4,3,1,2,2,1,42,1,2,2,1,3,4,14,4,3,1,2,2,1,42,1,
%U A040449 2,2,1,3,4,14,4,3,1,2,2,1,42,1,2,2,1,3,4,14,4,3,1,2,2,1,42,1,2
%N A040449 Continued fraction for sqrt(471).
%H A040449 Index entries for continued fractions for constants
%p A040449 with(numtheory): Digits:=300: convert(evalf(sqrt(471)),confrac);
%K A040449 nonn,cofr,easy
%O A040449 0,1
%A A040449 njas
%I A040448
%S A040448 21,1,2,8,2,1,42,1,2,8,2,1,42,1,2,8,2,1,42,1,2,8,2,1,42,1,2,8,2,
%T A040448 1,42,1,2,8,2,1,42,1,2,8,2,1,42,1,2,8,2,1,42,1,2,8,2,1,42,1,2,8,
%U A040448 2,1,42,1,2,8,2,1,42,1,2,8,2,1,42,1,2,8,2,1,42,1,2,8,2,1,42,1,2
%N A040448 Continued fraction for sqrt(470).
%H A040448 Index entries for continued fractions for constants
%p A040448 with(numtheory): Digits:=300: convert(evalf(sqrt(470)),confrac);
%K A040448 nonn,cofr,easy
%O A040448 0,1
%A A040448 njas
%I A040453
%S A040453 21,1,3,1,6,2,6,1,3,1,42,1,3,1,6,2,6,1,3,1,42,1,3,1,6,2,6,1,3,1,
%T A040453 42,1,3,1,6,2,6,1,3,1,42,1,3,1,6,2,6,1,3,1,42,1,3,1,6,2,6,1,3,1,
%U A040453 42,1,3,1,6,2,6,1,3,1,42,1,3,1,6,2,6,1,3,1,42,1,3,1,6,2,6,1,3,1
%N A040453 Continued fraction for sqrt(475).
%H A040453 Index entries for continued fractions for constants
%p A040453 with(numtheory): Digits:=300: convert(evalf(sqrt(475)),confrac);
%K A040453 nonn,cofr,easy
%O A040453 0,1
%A A040453 njas
%I A040452
%S A040452 21,1,3,2,1,1,1,6,1,1,1,2,3,1,42,1,3,2,1,1,1,6,1,1,1,2,3,1,42,1,
%T A040452 3,2,1,1,1,6,1,1,1,2,3,1,42,1,3,2,1,1,1,6,1,1,1,2,3,1,42,1,3,2,
%U A040452 1,1,1,6,1,1,1,2,3,1,42,1,3,2,1,1,1,6,1,1,1,2,3,1,42,1,3,2,1,1
%N A040452 Continued fraction for sqrt(474).
%H A040452 Index entries for continued fractions for constants
%p A040452 with(numtheory): Digits:=300: convert(evalf(sqrt(474)),confrac);
%K A040452 nonn,cofr,easy
%O A040452 0,1
%A A040452 njas
%I A040454
%S A040454 21,1,4,2,10,2,4,1,42,1,4,2,10,2,4,1,42,1,4,2,10,2,4,1,42,1,4,2,
%T A040454 10,2,4,1,42,1,4,2,10,2,4,1,42,1,4,2,10,2,4,1,42,1,4,2,10,2,4,1,
%U A040454 42,1,4,2,10,2,4,1,42,1,4,2,10,2,4,1,42,1,4,2,10,2,4,1,42,1,4,2
%N A040454 Continued fraction for sqrt(476).
%H A040454 Index entries for continued fractions for constants
%p A040454 with(numtheory): Digits:=300: convert(evalf(sqrt(476)),confrac);
%K A040454 nonn,cofr,easy
%O A040454 0,1
%A A040454 njas
%I A040455
%S A040455 21,1,5,3,1,4,10,1,2,2,4,2,2,1,10,4,1,3,5,1,42,1,5,3,1,4,10,1,2,
%T A040455 2,4,2,2,1,10,4,1,3,5,1,42,1,5,3,1,4,10,1,2,2,4,2,2,1,10,4,1,3,
%U A040455 5,1,42,1,5,3,1,4,10,1,2,2,4,2,2,1,10,4,1,3,5,1,42,1,5,3,1,4,10
%N A040455 Continued fraction for sqrt(477).
%H A040455 Index entries for continued fractions for constants
%p A040455 with(numtheory): Digits:=300: convert(evalf(sqrt(477)),confrac);
%K A040455 nonn,cofr,easy
%O A040455 0,1
%A A040455 njas
%I A040456
%S A040456 21,1,6,3,4,1,1,5,1,2,3,1,1,1,1,1,13,1,20,1,13,1,1,1,1,1,3,2,1,
%T A040456 5,1,1,4,3,6,1,42,1,6,3,4,1,1,5,1,2,3,1,1,1,1,1,13,1,20,1,13,1,
%U A040456 1,1,1,1,3,2,1,5,1,1,4,3,6,1,42,1,6,3,4,1,1,5,1,2,3,1,1,1,1,1,13
%N A040456 Continued fraction for sqrt(478).
%H A040456 Index entries for continued fractions for constants
%p A040456 with(numtheory): Digits:=300: convert(evalf(sqrt(478)),confrac);
%K A040456 nonn,cofr,easy
%O A040456 0,1
%A A040456 njas
%I A040457
%S A040457 21,1,7,1,3,2,21,2,3,1,7,1,42,1,7,1,3,2,21,2,3,1,7,1,42,1,7,1,3,
%T A040457 2,21,2,3,1,7,1,42,1,7,1,3,2,21,2,3,1,7,1,42,1,7,1,3,2,21,2,3,1,
%U A040457 7,1,42,1,7,1,3,2,21,2,3,1,7,1,42,1,7,1,3,2,21,2,3,1,7,1,42,1,7
%N A040457 Continued fraction for sqrt(479).
%H A040457 Index entries for continued fractions for constants
%p A040457 with(numtheory): Digits:=300: convert(evalf(sqrt(479)),confrac);
%K A040457 nonn,cofr,easy
%O A040457 0,1
%A A040457 njas
%I A040458
%S A040458 21,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,
%T A040458 42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,
%U A040458 42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1,42,1,9,1
%N A040458 Continued fraction for sqrt(480).
%H A040458 Index entries for continued fractions for constants
%p A040458 with(numtheory): Digits:=300: convert(evalf(sqrt(480)),confrac);
%K A040458 nonn,cofr,easy
%O A040458 0,1
%A A040458 njas
%I A040459
%S A040459 21,1,13,1,1,1,4,4,1,1,1,13,1,42,1,13,1,1,1,4,4,1,1,1,13,1,42,1,
%T A040459 13,1,1,1,4,4,1,1,1,13,1,42,1,13,1,1,1,4,4,1,1,1,13,1,42,1,13,1,
%U A040459 1,1,4,4,1,1,1,13,1,42,1,13,1,1,1,4,4,1,1,1,13,1,42,1,13,1,1,1
%N A040459 Continued fraction for sqrt(481).
%H A040459 Index entries for continued fractions for constants
%p A040459 with(numtheory): Digits:=300: convert(evalf(sqrt(481)),confrac);
%K A040459 nonn,cofr,easy
%O A040459 0,1
%A A040459 njas
%I A040460
%S A040460 21,1,20,1,42,1,20,1,42,1,20,1,42,1,20,1,42,1,20,1,42,1,20,1,42,
%T A040460 1,20,1,42,1,20,1,42,1,20,1,42,1,20,1,42,1,20,1,42,1,20,1,42,1,
%U A040460 20,1,42,1,20,1,42,1,20,1,42,1,20,1,42,1,20,1,42,1,20,1,42,1,20
%N A040460 Continued fraction for sqrt(482).
%H A040460 Index entries for continued fractions for constants
%p A040460 with(numtheory): Digits:=300: convert(evalf(sqrt(482)),confrac);
%K A040460 nonn,cofr,easy
%O A040460 0,1
%A A040460 njas
%I A040461
%S A040461 21,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,
%T A040461 1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,
%U A040461 42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42,1,42
%N A040461 Continued fraction for sqrt(483).
%H A040461 Index entries for continued fractions for constants
%p A040461 with(numtheory): Digits:=300: convert(evalf(sqrt(483)),confrac);
%K A040461 nonn,cofr,easy
%O A040461 0,1
%A A040461 njas
%I A013530
%S A013530 1,0,21,1,1958,206,3190,123635,827754000,185433755,2409296136229,
%T A013530 587755907572,20386280235589,3504664356222853
%V A013530 1,0,21,-1,1958,-206,3190,-123635,827754000,-185433755,
%W A013530 2409296136229,-587755907572,20386280235589,-3504664356222853
%N A013530 sin(cosec(x)-cosech(x))=x+21/5!*x^5-1/7!*x^7+1958/9!*x^9-206/11!*x^11...
%K A013530 sign,done
%O A013530 0,3
%A A013530 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A013531
%S A013531 1,0,21,1,2772,281,8000,255521,4367718461,705140399,27918839266101,
%T A013531 4781541644580,524472263000036,67827241362679118
%N A013531 arcsin(cosec(x)-cosech(x))=x+21/5!*x^5+1/7!*x^7+2772/9!*x^9...
%K A013531 nonn
%O A013531 0,3
%A A013531 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A040435
%S A040435 21,2,1,1,1,5,2,13,1,3,1,4,1,1,4,1,3,1,13,2,5,1,1,1,2,42,2,1,1,
%T A040435 1,5,2,13,1,3,1,4,1,1,4,1,3,1,13,2,5,1,1,1,2,42,2,1,1,1,5,2,13,
%U A040435 1,3,1,4,1,1,4,1,3,1,13,2,5,1,1,1,2,42,2,1,1,1,5,2,13,1,3,1,4,1
%N A040435 Continued fraction for sqrt(457).
%H A040435 Index entries for continued fractions for constants
%p A040435 with(numtheory): Digits:=300: convert(evalf(sqrt(457)),confrac);
%K A040435 nonn,cofr,easy
%O A040435 0,1
%A A040435 njas
%I A040434
%S A040434 21,2,1,4,1,2,42,2,1,4,1,2,42,2,1,4,1,2,42,2,1,4,1,2,42,2,1,4,1,
%T A040434 2,42,2,1,4,1,2,42,2,1,4,1,2,42,2,1,4,1,2,42,2,1,4,1,2,42,2,1,4,
%U A040434 1,2,42,2,1,4,1,2,42,2,1,4,1,2,42,2,1,4,1,2,42,2,1,4,1,2,42,2,1
%N A040434 Continued fraction for sqrt(456).
%H A040434 Index entries for continued fractions for constants
%p A040434 with(numtheory): Digits:=300: convert(evalf(sqrt(456)),confrac);
%K A040434 nonn,cofr,easy
%O A040434 0,1
%A A040434 njas
%I A040437
%S A040437 21,2,2,1,4,21,4,1,2,2,42,2,2,1,4,21,4,1,2,2,42,2,2,1,4,21,4,1,
%T A040437 2,2,42,2,2,1,4,21,4,1,2,2,42,2,2,1,4,21,4,1,2,2,42,2,2,1,4,21,
%U A040437 4,1,2,2,42,2,2,1,4,21,4,1,2,2,42,2,2,1,4,21,4,1,2,2,42,2,2,1,4
%N A040437 Continued fraction for sqrt(459).
%H A040437 Index entries for continued fractions for constants
%p A040437 with(numtheory): Digits:=300: convert(evalf(sqrt(459)),confrac);
%K A040437 nonn,cofr,easy
%O A040437 0,1
%A A040437 njas
%I A040436
%S A040436 21,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,
%T A040436 42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,
%U A040436 42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2,42,2,2
%N A040436 Continued fraction for sqrt(458).
%H A040436 Index entries for continued fractions for constants
%p A040436 with(numtheory): Digits:=300: convert(evalf(sqrt(458)),confrac);
%K A040436 nonn,cofr,easy
%O A040436 0,1
%A A040436 njas
%I A040438
%S A040438 21,2,4,3,1,2,10,2,1,3,4,2,42,2,4,3,1,2,10,2,1,3,4,2,42,2,4,3,1,
%T A040438 2,10,2,1,3,4,2,42,2,4,3,1,2,10,2,1,3,4,2,42,2,4,3,1,2,10,2,1,3,
%U A040438 4,2,42,2,4,3,1,2,10,2,1,3,4,2,42,2,4,3,1,2,10,2,1,3,4,2,42,2,4
%N A040438 Continued fraction for sqrt(460).
%H A040438 Index entries for continued fractions for constants
%p A040438 with(numtheory): Digits:=300: convert(evalf(sqrt(460)),confrac);
%K A040438 nonn,cofr,easy
%O A040438 0,1
%A A040438 njas
%I A040439
%S A040439 21,2,8,10,1,1,1,1,1,1,1,1,10,8,2,42,2,8,10,1,1,1,1,1,1,1,1,10,
%T A040439 8,2,42,2,8,10,1,1,1,1,1,1,1,1,10,8,2,42,2,8,10,1,1,1,1,1,1,1,1,
%U A040439 10,8,2,42,2,8,10,1,1,1,1,1,1,1,1,10,8,2,42,2,8,10,1,1,1,1,1,1
%N A040439 Continued fraction for sqrt(461).
%H A040439 Index entries for continued fractions for constants
%p A040439 with(numtheory): Digits:=300: convert(evalf(sqrt(461)),confrac);
%K A040439 nonn,cofr,easy
%O A040439 0,1
%A A040439 njas
%I A040440
%S A040440 21,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,
%T A040440 2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,
%U A040440 42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42,2,42
%N A040440 Continued fraction for sqrt(462).
%H A040440 Index entries for sequences related to covering codes
%H A040440 Index entries for continued fractions for constants
%p A040440 with(numtheory): Digits:=300: convert(evalf(sqrt(462)),confrac);
%K A040440 nonn,cofr,easy
%O A040440 0,1
%A A040440 njas
%I A013532
%S A013532 1,0,21,2,3399,499,10867,386756,5560448040,851066318,29099101102308,
%T A013532 4367412770048,411109225002823,44984491359492820
%N A013532 tan(cosec(x)-cosech(x))=x+21/5!*x^5+2/7!*x^7+3399/9!*x^9+499/11!*x^11...
%K A013532 nonn
%O A013532 0,3
%A A013532 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A040431
%S A040431 21,3,1,1,10,14,10,1,1,3,42,3,1,1,10,14,10,1,1,3,42,3,1,1,10,14,
%T A040431 10,1,1,3,42,3,1,1,10,14,10,1,1,3,42,3,1,1,10,14,10,1,1,3,42,3,
%U A040431 1,1,10,14,10,1,1,3,42,3,1,1,10,14,10,1,1,3,42,3,1,1,10,14,10,1
%N A040431 Continued fraction for sqrt(453).
%H A040431 Index entries for continued fractions for constants
%p A040431 with(numtheory): Digits:=300: convert(evalf(sqrt(453)),confrac);
%K A040431 nonn,cofr,easy
%O A040431 0,1
%A A040431 njas
%I A040430
%S A040430 21,3,1,5,3,10,3,5,1,3,42,3,1,5,3,10,3,5,1,3,42,3,1,5,3,10,3,5,
%T A040430 1,3,42,3,1,5,3,10,3,5,1,3,42,3,1,5,3,10,3,5,1,3,42,3,1,5,3,10,
%U A040430 3,5,1,3,42,3,1,5,3,10,3,5,1,3,42,3,1,5,3,10,3,5,1,3,42,3,1,5,3
%N A040430 Continued fraction for sqrt(452).
%H A040430 Index entries for continued fractions for constants
%p A040430 with(numtheory): Digits:=300: convert(evalf(sqrt(452)),confrac);
%K A040430 nonn,cofr,easy
%O A040430 0,1
%A A040430 njas
%I A040432
%S A040432 21,3,3,1,13,2,3,2,1,1,4,6,1,7,1,1,1,20,1,1,1,7,1,6,4,1,1,2,3,2,
%T A040432 13,1,3,3,42,3,3,1,13,2,3,2,1,1,4,6,1,7,1,1,1,20,1,1,1,7,1,6,4,
%U A040432 1,1,2,3,2,13,1,3,3,42,3,3,1,13,2,3,2,1,1,4,6,1,7,1,1,1,20,1,1
%N A040432 Continued fraction for sqrt(454).
%H A040432 Index entries for continued fractions for constants
%p A040432 with(numtheory): Digits:=300: convert(evalf(sqrt(454)),confrac);
%K A040432 nonn,cofr,easy
%O A040432 0,1
%A A040432 njas
%I A035419
%S A035419 21,3,4,8,105,5,31,5,11,9,5,44,3,9,423,5,36,12,5,5,16,3,116,7,13,9,9,5,
%T A035419 54,16,3,9
%N A035419 Related to Rogers-Ramanujan Identities.
%D A035419 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
%D A035419 Contact author at address below [ Note: this form of description is not accepted! - NJAS ]
%Y A035419 Cf. A003106, A035409, A035416, A035417, A035418, A035420.
%K A035419 nonn,more,part
%O A035419 1,1
%A A035419 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A040433
%S A040433 21,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,
%T A040433 3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,
%U A040433 42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42,3,42
%N A040433 Continued fraction for sqrt(455).
%H A040433 Index entries for continued fractions for constants
%p A040433 with(numtheory): Digits:=300: convert(evalf(sqrt(455)),confrac);
%K A040433 nonn,cofr,easy
%O A040433 0,1
%A A040433 njas
%I A018855
%S A018855 1,21,3,45,55,6,78,820,91,10,1128,120,136,1431,15,1653,171,1830,190,2016,
%T A018855 21,2211,231,2415,253,2628,276,28,2926,300,3160,325,3321,3403,351,36,378,
%U A018855 3828,3916,406,4186,4278,435,4465,45,465,4753,4851,496,5050,5151,528
%N A018855 Smallest triangular number that begins with n.
%Y A018855 Cf. A018801.
%K A018855 nonn,base
%O A018855 1,2
%A A018855 dww
%I A040428
%S A040428 21,4,1,2,4,2,1,4,42,4,1,2,4,2,1,4,42,4,1,2,4,2,1,4,42,4,1,2,4,
%T A040428 2,1,4,42,4,1,2,4,2,1,4,42,4,1,2,4,2,1,4,42,4,1,2,4,2,1,4,42,4,
%U A040428 1,2,4,2,1,4,42,4,1,2,4,2,1,4,42,4,1,2,4,2,1,4,42,4,1,2,4,2,1,4
%N A040428 Continued fraction for sqrt(450).
%H A040428 Index entries for continued fractions for constants
%p A040428 with(numtheory): Digits:=300: convert(evalf(sqrt(450)),confrac);
%K A040428 nonn,cofr,easy
%O A040428 0,1
%A A040428 njas
%I A040429
%S A040429 21,4,4,2,8,21,8,2,4,4,42,4,4,2,8,21,8,2,4,4,42,4,4,2,8,21,8,2,
%T A040429 4,4,42,4,4,2,8,21,8,2,4,4,42,4,4,2,8,21,8,2,4,4,42,4,4,2,8,21,
%U A040429 8,2,4,4,42,4,4,2,8,21,8,2,4,4,42,4,4,2,8,21,8,2,4,4,42,4,4,2,8
%N A040429 Continued fraction for sqrt(451).
%H A040429 Index entries for continued fractions for constants
%p A040429 with(numtheory): Digits:=300: convert(evalf(sqrt(451)),confrac);
%K A040429 nonn,cofr,easy
%O A040429 0,1
%A A040429 njas
%I A034082
%S A034082 21,4,6,8,12,18,26,39,58,87,130,195,292,438,657,986,1478,2217,3326,
%T A034082 4988,7482,11223,16835,25252,37877,56816,85223,127835,191752,287627,
%U A034082 431440,647160,970740,1456110,2184165,3276247,4914370,7371555,11057333
%N A034082 Decimal part of n-th root of a(n) starts with digit 5.
%F A034082 For n>2: a(n) =ceiling[(3/2)^n] =A002379(n)+1 - Henry Bottomley (se16@btinternet.com), May 02 2001
%e A034082 a(25)=25252 -> 25252^(1/25)=1.{5}000019762083...
%Y A034082 Cf. A034062, A034072, A002379, A061418, A061419.
%K A034082 nonn,base
%O A034082 2,1
%A A034082 Patrick De Geest (pdg@worldofnumbers.com), Sep 1998.
%I A040427
%S A040427 21,5,3,1,1,1,7,1,5,5,1,7,1,1,1,3,5,42,5,3,1,1,1,7,1,5,5,1,7,1,
%T A040427 1,1,3,5,42,5,3,1,1,1,7,1,5,5,1,7,1,1,1,3,5,42,5,3,1,1,1,7,1,5,
%U A040427 5,1,7,1,1,1,3,5,42,5,3,1,1,1,7,1,5,5,1,7,1,1,1,3,5,42,5,3,1,1
%N A040427 Continued fraction for sqrt(449).
%H A040427 Index entries for continued fractions for constants
%p A040427 with(numtheory): Digits:=300: convert(evalf(sqrt(449)),confrac);
%K A040427 nonn,cofr,easy
%O A040427 0,1
%A A040427 njas
%I A040426
%S A040426 21,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,
%T A040426 6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,
%U A040426 42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42,6,42
%N A040426 Continued fraction for sqrt(448).
%H A040426 Index entries for continued fractions for constants
%p A040426 with(numtheory): Digits:=300: convert(evalf(sqrt(448)),confrac);
%K A040426 nonn,cofr,easy
%O A040426 0,1
%A A040426 njas
%I A040425
%S A040425 21,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,
%T A040425 7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,
%U A040425 42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42,7,42
%N A040425 Continued fraction for sqrt(447).
%H A040425 Index entries for continued fractions for constants
%p A040425 with(numtheory): Digits:=300: convert(evalf(sqrt(447)),confrac);
%K A040425 nonn,cofr,easy
%O A040425 0,1
%A A040425 njas
%I A040424
%S A040424 21,8,2,2,1,3,1,1,20,1,1,3,1,2,2,8,42,8,2,2,1,3,1,1,20,1,1,3,1,
%T A040424 2,2,8,42,8,2,2,1,3,1,1,20,1,1,3,1,2,2,8,42,8,2,2,1,3,1,1,20,1,
%U A040424 1,3,1,2,2,8,42,8,2,2,1,3,1,1,20,1,1,3,1,2,2,8,42,8,2,2,1,3,1,1
%N A040424 Continued fraction for sqrt(446).
%H A040424 Index entries for continued fractions for constants
%p A040424 with(numtheory): Digits:=300: convert(evalf(sqrt(446)),confrac);
%K A040424 nonn,cofr,easy
%O A040424 0,1
%A A040424 njas
%I A040423
%S A040423 21,10,1,1,10,42,10,1,1,10,42,10,1,1,10,42,10,1,1,10,42,10,1,1,
%T A040423 10,42,10,1,1,10,42,10,1,1,10,42,10,1,1,10,42,10,1,1,10,42,10,1,
%U A040423 1,10,42,10,1,1,10,42,10,1,1,10,42,10,1,1,10,42,10,1,1,10,42,10
%N A040423 Continued fraction for sqrt(445).
%H A040423 Index entries for continued fractions for constants
%p A040423 with(numtheory): Digits:=300: convert(evalf(sqrt(445)),confrac);
%K A040423 nonn,cofr,easy
%O A040423 0,1
%A A040423 njas
%I A033341
%S A033341 21,10,7,5,4,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A033341 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A033341 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A033341 [ 21/n ].
%K A033341 easy,nonn
%O A033341 1,1
%A A033341 Jeff Burch (jmburch@osprey.smcm.edu)
%I A040422
%S A040422 21,14,42,14,42,14,42,14,42,14,42,14,42,14,42,14,42,14,42,14,42,
%T A040422 14,42,14,42,14,42,14,42,14,42,14,42,14,42,14,42,14,42,14,42,14,
%U A040422 42,14,42,14,42,14,42,14,42,14,42,14,42,14,42,14,42,14,42,14,42
%N A040422 Continued fraction for sqrt(444).
%H A040422 Index entries for continued fractions for constants
%p A040422 with(numtheory): Digits:=300: convert(evalf(sqrt(444)),confrac);
%K A040422 nonn,cofr,easy
%O A040422 0,1
%A A040422 njas
%I A048933
%S A048933 21,15,35,30,21,27,80,201,260,210,204,150,135,158,152,161,167,141,201,
%T A048933 231,281,152,231,204,246,251,201,261,140,179,311,323,315,317,231,351,
%U A048933 215,146,350,351,317,156,300,251,261,356,240,269,165,176,396,221,231
%N A048933 Smaller vampire number fangs corresponding to A014575.
%D A048933 C. A. Pickover, "Vampire Numbers." Ch. 30 in Keys to Infinity. New York: Wiley, pp. 227-231, 1995.
%H A048933 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A048933 Cf. A014575, A048934, ..., A048939.
%K A048933 nonn
%O A048933 1,1
%A A048933 Eric W. Weisstein (eric@weisstein.com)
%I A022977
%S A022977 21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,
%T A022977 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,
%U A022977 22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39
%V A022977 21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,
%W A022977 0,-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,-11,-12,-13,-14,-15,-16,-17,-18,-19,-20,-21,
%X A022977 -22,-23,-24,-25,-26,-27,-28,-29,-30,-31,-32,-33,-34,-35,-36,-37,-38,-39
%N A022977 21-n.
%K A022977 sign,done
%O A022977 0,1
%A A022977 njas
%I A023463
%S A023463 21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,
%T A023463 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,
%U A023463 22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39
%V A023463 -21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,
%W A023463 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,
%X A023463 22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39
%N A023463 n-21.
%K A023463 sign,done
%O A023463 0,1
%A A023463 njas
%I A004462
%S A004462 21,20,23,22,17,16,19,18,29,28,31,30,25,24,27,26,5,4,7,
%T A004462 6,1,0,3,2,13,12,15,14,9,8,11,10,53,52,55,54,49,48,51,50,
%U A004462 61,60,63,62,57,56,59,58,37,36,39,38,33,32,35,34,45,44
%N A004462 Nimsum n + 21.
%D A004462 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
%D A004462 J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
%H A004462 Index entries for sequences related to Nim-sums
%K A004462 nonn
%O A004462 0,1
%A A004462 njas
%I A010860
%S A010860 21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,
%T A010860 21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,
%U A010860 21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21
%N A010860 Constant sequence.
%K A010860 nonn
%O A010860 0,1
%A A010860 njas
%I A040421
%S A040421 21,21,42,21,42,21,42,21,42,21,42,21,42,21,42,21,42,21,42,21,42,
%T A040421 21,42,21,42,21,42,21,42,21,42,21,42,21,42,21,42,21,42,21,42,21,
%U A040421 42,21,42,21,42,21,42,21,42,21,42,21,42,21,42,21,42,21,42,21,42
%N A040421 Continued fraction for sqrt(443).
%H A040421 Index entries for continued fractions for constants
%p A040421 with(numtheory): Digits:=300: convert(evalf(sqrt(443)),confrac);
%K A040421 nonn,cofr,easy
%O A040421 0,1
%A A040421 njas
%I A022355
%S A022355 0,21,21,42,63,105,168,273,441,714,1155,1869,3024,4893,
%T A022355 7917,12810,20727,33537,54264,87801,142065,229866,371931,
%U A022355 601797,973728,1575525,2549253,4124778,6674031,10798809
%N A022355 Fibonacci sequence beginning 0 21.
%K A022355 nonn
%O A022355 0,2
%A A022355 njas
%I A048245
%S A048245 0,0,0,0,0,21,21,58,23,15
%N A048245 Number of distinct winning tic-tac-toe positions after n plays, up to rotation and reflection.
%C A048245 As in chess, a "play" is an action by a single player, a "move" is two contiguous plays by each opponent.
%H A048245 Index entries for sequences related to tic-tac-toe
%Y A048245 A008907.
%K A048245 fini,full,nonn
%O A048245 0,6
%A A048245 Robert W. Pratt (rpratt@email.unc.edu)
%I A056485
%S A056485 0,0,0,0,0,0,0,0,0,0,1,1,21,21,266,266,2646,2646,22827,22827,179487,
%T A056485 179486,1323652,1323651,9321312,9321291,63436373,63436352,420693273,
%U A056485 420693007,2734926558
%N A056485 Primitive (aperiodic) palindromic structures using exactly six different characters.
%C A056485 Permuting the characters will not change the structure.
%D A056485 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%F A056485 A056480(n)-A056479(n).
%Y A056485 Cf. A056467.
%K A056485 nonn
%O A056485 1,13
%A A056485 Marks R. Nester (nesterm@qfri1.se2.dpi.qld.gov.au)
%I A056475
%S A056475 0,0,0,0,0,0,0,0,0,0,1,1,21,21,266,266,2646,2646,22827,22827,179487,
%T A056475 179487,1323652,1323652,9321312,9321312,63436373,63436373,420693273,
%U A056475 420693273,2734926558
%N A056475 Palindromic structures using exactly six different characters.
%C A056475 Permuting the characters will not change the structure.
%D A056475 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%F A056475 stirling2( [(n+1)/2], 6).
%Y A056475 Cf. A000770, A056471.
%K A056475 nonn
%O A056475 1,13
%A A056475 Marks R. Nester (nesterm@qfri1.se2.dpi.qld.gov.au)
%I A004510
%S A004510 21,22,23,24,25,26,18,19,20,3,4,5,6,7,8,0,1,2,12,13,14,
%T A004510 15,16,17,9,10,11,48,49,50,51,52,53,45,46,47,30,31,32,33,
%U A004510 34,35,27,28,29,39,40,41,42,43,44,36,37,38,75,76,77,78
%N A004510 Tersum n + 21.
%F A004510 Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g. 5 + 8 = "21" + "22" = "10" = 1.
%K A004510 nonn
%O A004510 0,1
%A A004510 njas
%I A022391
%S A022391 1,21,22,43,65,108,173,281,454,735,1189,1924,3113,5037,
%T A022391 8150,13187,21337,34524,55861,90385,146246,236631,382877,
%U A022391 619508,1002385,1621893,2624278,4246171,6870449,11116620
%N A022391 Fibonacci sequence beginning 1 21.
%K A022391 nonn
%O A022391 0,2
%A A022391 njas
%I A041890
%S A041890 21,22,43,65,108,389,1275,27164,82767,275465,358232,633697,
%T A041890 991929,1625626,69268221,70893847,140162068,211055915,351217983,
%U A041890 1264709864,4145347575,88317008939,269096374392,895606132115
%N A041890 Numerators of continued fraction convergents to sqrt(467).
%Y A041890 Cf. A041891.
%K A041890 nonn,cofr,easy
%O A041890 0,1
%A A041890 njas
%I A041892
%S A041892 21,22,43,65,173,238,411,649,27669,28318,55987,84305,224597,
%T A041892 308902,533499,842401,35914341,36756742,72671083,109427825,
%U A041892 291526733,400954558,692481291,1093435849,46616786949,47710222798
%N A041892 Numerators of continued fraction convergents to sqrt(468).
%Y A041892 Cf. A041893.
%K A041892 nonn,cofr,easy
%O A041892 0,1
%A A041892 njas
%I A041894
%S A041894 21,22,43,65,693,4223,42923,47146,90069,137215,5853099,
%T A041894 5990314,11843413,17833727,190180683,1158917825,11779358933,
%U A041894 12938276758,24717635691,37655912449,1606265958549,1643921870998
%N A041894 Numerators of continued fraction convergents to sqrt(469).
%Y A041894 Cf. A041895.
%K A041894 nonn,cofr,easy
%O A041894 0,1
%A A041894 njas
%I A041888
%S A041888 21,22,43,108,259,367,993,5332,6325,24307,30632,636947,
%T A041888 667579,2639684,3307263,19175999,41659261,60835260,163329781,
%U A041888 387494822,550824603,938319425,39960240453,40898559878
%N A041888 Numerators of continued fraction convergents to sqrt(466).
%Y A041888 Cf. A041889.
%K A041888 nonn,cofr,easy
%O A041888 0,1
%A A041888 njas
%I A041886
%S A041886 21,22,43,151,345,841,2027,6922,8949,15871,675531,691402,
%T A041886 1366933,4792201,10951335,26694871,64341077,219718102,284059179,
%U A041886 503777281,21442704981,21946482262,43389187243,152114043991
%N A041886 Numerators of continued fraction convergents to sqrt(465).
%Y A041886 Cf. A041887.
%K A041886 nonn,cofr,easy
%O A041886 0,1
%A A041886 njas
%I A041884
%S A041884 21,22,43,237,280,517,797,4502,5299,9801,416941,426742,
%T A041884 843683,4645157,5488840,10133997,15622837,88248182,103871019,
%U A041884 192119201,8172877461,8364996662,16537874123,91054367277
%N A041884 Numerators of continued fraction convergents to sqrt(464).
%Y A041884 Cf. A041885.
%K A041884 nonn,cofr,easy
%O A041884 0,1
%A A041884 njas
%I A041882
%S A041882 21,22,43,581,624,3701,15428,19129,34557,53686,88243,230172,
%T A041882 548587,3521694,4070281,15732537,334453558,1019093211,1353546769,
%U A041882 9140373825,19634294419,48408962663,68043257082,116452219745
%N A041882 Numerators of continued fraction convergents to sqrt(463).
%Y A041882 Cf. A041883.
%K A041882 nonn,cofr,easy
%O A041882 0,1
%A A041882 njas
%I A041900
%S A041900 21,22,65,87,152,239,1108,5779,24224,30003,54227,84230,
%T A041900 222687,306917,13113201,13420118,39953437,53373555,93326992,
%U A041900 146700547,680129180,3547346447,14869514968,18416861415
%N A041900 Numerators of continued fraction convergents to sqrt(472).
%Y A041900 Cf. A041901.
%K A041900 nonn,cofr,easy
%O A041900 0,1
%A A041900 njas
%I A041902
%S A041902 21,22,65,87,3719,3806,11331,15137,647085,662222,1971529,
%T A041902 2633751,112589071,115222822,343034715,458257537,19589851269,
%U A041902 20048108806,59686068881,79734177687,3408521531735,3488255709422
%N A041902 Numerators of continued fraction convergents to sqrt(473).
%Y A041902 Cf. A041903.
%K A041902 nonn,cofr,easy
%O A041902 0,1
%A A041902 njas
%I A041898
%S A041898 21,22,65,152,217,803,3429,48809,198665,644804,843469,2331742,
%T A041898 5506953,7838695,334732143,342570838,1019873819,2382318476,
%U A041898 3402192295,12588895361,53757773739,765197727707,3114548684567
%N A041898 Numerators of continued fraction convergents to sqrt(471).
%Y A041898 Cf. A041899.
%K A041898 nonn,cofr,easy
%O A041898 0,1
%A A041898 njas
%I A041896
%S A041896 21,22,65,542,1149,1691,72171,73862,219895,1833022,3885939,
%T A041896 5718961,244082301,249801262,743684825,6199279862,13142244549,
%U A041896 19341524411,825486269811,844827794222,2515141858255,20965962660262
%N A041896 Numerators of continued fraction convergents to sqrt(470).
%Y A041896 Cf. A041897.
%K A041896 nonn,cofr,easy
%O A041896 0,1
%A A041896 njas
%I A041906
%S A041906 21,22,87,109,741,1591,10287,11878,45921,57799,2473479,
%T A041906 2531278,10067313,12598591,85658859,183916309,1189156713,
%U A041906 1373073022,5308375779,6681448801,285929225421,292610674222
%N A041906 Numerators of continued fraction convergents to sqrt(475).
%Y A041906 Cf. A041907.
%K A041906 nonn,cofr,easy
%O A041906 0,1
%A A041906 njas
%I A041904
%S A041904 21,22,87,196,283,479,762,5051,5813,10864,16677,44218,149331,
%T A041904 193549,8278389,8471938,33694203,75860344,109554547,185414891,
%U A041904 294969438,1955231519,2250200957,4205432476,6455633433
%N A041904 Numerators of continued fraction convergents to sqrt(474).
%Y A041904 Cf. A041905.
%K A041904 nonn,cofr,easy
%O A041904 0,1
%A A041904 njas
%I A041908
%S A041908 21,22,109,240,2509,5258,23541,28799,1233099,1261898,6280691,
%T A041908 13823280,144513491,302850262,1355914539,1658764801,71024036181,
%U A041908 72682800982,361755240109,796193281200,8323688052109,17443569385418
%N A041908 Numerators of continued fraction convergents to sqrt(476).
%Y A041908 Cf. A041909.
%K A041908 nonn,cofr,easy
%O A041908 0,1
%A A041908 njas
%I A055662
%S A055662 0,1,21,22,122,120,110,111,2111,2112,2102,2100,2200,2201,2221,2222,
%T A055662 12222,12220,12210,12211,12011,12012,12002,12000,11000,11001,11021,
%U A055662 11022,11122,11120,11110,11111,211111,211112,211102,211100,211200
%N A055662 Successive positions in Tower of Hanoi (with three pegs {0,1,2}) where xyz means smallest disk is on peg z, second smallest is on peg y, third smallest on peg x, etc. and leading zeros indicate largest disks are all on peg 0.
%C A055662 Optimal for moving an even number of disks from peg 0 to peg 2 or an odd number from peg 0 to peg 1
%F A055662 a(n) =sum( 10^j * ((floor[(n/2^j+1)/2]*(-1)^j mod 3) ) with j=0 to floor[log2(n)]
%Y A055662 Cf. A001511, A055661.
%K A055662 nonn
%O A055662 0,3
%A A055662 Henry Bottomley (se16@btinternet.com), Jun 06 2000
%I A041910
%S A041910 21,22,131,415,546,2599,26536,29135,84806,198747,879794,
%T A041910 1958335,4796464,6754799,72344454,296132615,368477069,1401563822,
%U A041910 7376296179,8777860001,376046416221,384824276222,2300167797331
%N A041910 Numerators of continued fraction convergents to sqrt(477).
%Y A041910 Cf. A041911.
%K A041910 nonn,cofr,easy
%O A041910 0,1
%A A041910 njas
%I A041912
%S A041912 21,22,153,481,2077,2558,4635,25733,30368,86469,289775,
%T A041912 376244,666019,1042263,1708282,2750545,37465367,40215912,
%U A041912 841783607,881999519,12307777354,13189776873,25497554227
%N A041912 Numerators of continued fraction convergents to sqrt(478).
%Y A041912 Cf. A041913.
%K A041912 nonn,cofr,easy
%O A041912 0,1
%A A041912 njas
%I A041914
%S A041914 21,22,175,197,766,1729,37075,75879,264712,340591,2648849,
%T A041914 2989440,128205329,131194769,1046568712,1177763481,4579859155,
%U A041914 10337481791,221666976766,453671435323,1582681282735,2036352718058
%N A041914 Numerators of continued fraction convergents to sqrt(479).
%Y A041914 Cf. A041915.
%K A041914 nonn,cofr,easy
%O A041914 0,1
%A A041914 njas
%I A041916
%S A041916 21,22,219,241,10341,10582,105579,116161,4984341,5100502,
%T A041916 50888859,55989361,2402442021,2458431382,24528324459,26986755841,
%U A041916 1157972069781,1184958825622,11822601500379,13007560326001
%N A041916 Numerators of continued fraction convergents to sqrt(480).
%Y A041916 Cf. A041917.
%K A041916 nonn,cofr,easy
%O A041916 0,1
%A A041916 njas
%I A041918
%S A041918 21,22,307,329,636,965,4496,18949,23445,42394,65839,898301,
%T A041918 964140,41392181,42356321,592024354,634380675,1226405029,
%U A041918 1860785704,8669547845,36538977084,45208524929,81747502013
%N A041918 Numerators of continued fraction convergents to sqrt(481).
%Y A041918 Cf. A041919.
%K A041918 nonn,cofr,easy
%O A041918 0,1
%A A041918 njas
%I A041219
%S A041219 1,1,21,22,461,483,10121,10604,222201,232805,4878301,5111106,
%T A041219 107100421,112211527,2351330961,2463542488,51622180721,
%U A041219 54085723209,1133336644901,1187422368110,24881784007101
%N A041219 Denominators of continued fraction convergents to sqrt(120).
%Y A041219 Cf. A041218.
%K A041219 nonn,cofr,easy
%O A041219 0,3
%A A041219 njas
%I A041920
%S A041920 21,22,461,483,20747,21230,445347,466577,20041581,20508158,
%T A041920 430204741,450712899,19360146499,19810859398,415577334459,
%U A041920 435388193857,18701881476453,19137269670310,401447274882653
%N A041920 Numerators of continued fraction convergents to sqrt(482).
%Y A041920 Cf. A041921.
%K A041920 nonn,cofr,easy
%O A041920 0,1
%A A041920 njas
%I A041921
%S A041921 1,1,21,22,945,967,20285,21252,912869,934121,19595289,20529410,
%T A041921 881830509,902359919,18929028889,19831388808,851847358825,
%U A041921 871678747633,18285422311485,19157101059118,822883666794441
%N A041921 Denominators of continued fraction convergents to sqrt(482).
%Y A041921 Cf. A041920.
%K A041921 nonn,cofr,easy
%O A041921 0,3
%A A041921 njas
%I A041922
%S A041922 21,22,945,967,41559,42526,1827651,1870177,80375085,82245262,
%T A041922 3534676089,3616921351,155445372831,159062294182,6836061728475,
%U A041922 6995124022657,300631270680069,307626394702726,13220939848194561
%N A041922 Numerators of continued fraction convergents to sqrt(483).
%Y A041922 Cf. A041923.
%K A041922 nonn,cofr,easy
%O A041922 0,1
%A A041922 njas
%I A007929
%S A007929 21,23,25,27,29,41,43,45,47,49,61,63,65,67,69,81,83,85,87,89,101,103,105,
%T A007929 107,109,121,123,125,127,129,141,143,145,147,149,161,163,165,167,169,181,
%U A007929 183,185,187,189,201
%N A007929 Odd numbers containing an even digit.
%D A007929 F. Smarandache, "Only Problems, not Solutions!", Xiquan Publ., Phoenix-Chicago, 1993
%H A007929 M. L. Perez et al., eds., Smarandache Notions Journal
%K A007929 nonn,base
%O A007929 1,1
%A A007929 R. Muller
%I A066867
%S A066867 21,24,27,32,40,46,56,62,73,85,94,141,157,164,170,175,183,188,216,228,
%T A066867 234,237,261,265,268,293,300,317,331,339,349,355,359,369,376,379,386,
%U A066867 403,410,430,442,447,451,454,458,463,472,495,498
%N A066867 Powers of two having 7 as the fourth digit from the right.
%C A066867 A sequence of no importance apart from the reference, which attributes the solution of this to John von Neumann, beating a computer to the solution.
%D A066867 Sylvia Nasar, A Beautiful Mind (1998), p. 80.
%t A066867 Select[ Range[ 10, 500 ], IntegerDigits[ 2^# ][ [ -4 ] ] == 7 & ]
%K A066867 nonn
%O A066867 0,1
%A A066867 Harvey P. Dale (hpd1@nyu.edu), Jan 21 2002
%I A033267
%S A033267 21,24,30,33,40,42,45,48,57,60,70,72,78,85,88,93,102,112,130,133,
%T A033267 177,190,232,253
%N A033267 n such that every genus of binary quadratic forms of discriminant -4n consists of a single class, and the class number h(-4n) = 4.
%D A033267 G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.
%D A033267 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989, p. 60.
%K A033267 nonn,fini,full
%O A033267 0,1
%A A033267 njas
%I A020269
%S A020269 21,25,185,925,1541,1807,3281,3439,3781,4417,7081,8857,10609,11989,14089,
%T A020269 18721,22849,26599,26825,30361,30673,36019,56033,58501,60409,81367,88507,
%U A020269 100553,114211,114247,120205,125569,133141,142801,183721,186497,188057
%N A020269 Strong pseudoprimes to base 43.
%H A020269 Index entries for sequences related to pseudoprimes
%K A020269 nonn
%O A020269 1,1
%A A020269 dww
%I A035700
%S A035700 21,26,30,70,75,80,106,124,125,133,142,180,191,200,231,268,278,297,
%T A035700 298,322,336,339,340,342,350,351,353,358,365,374,412,415,449,465,494,
%U A035700 501,531,548,550,570,579,580,602,632,645,648,649,657,663,674,679,699
%N A035700 Number of partitions of n is a multiple of 12.
%Y A035700 Cf. A000041, A035701.
%K A035700 nonn,part
%O A035700 1,1
%A A035700 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A064507
%S A064507 21,27,29,61,67,111,151,191,201,203,223,241,313,319,331,351,373,397,
%T A064507 403,409,461,463,481,553,559,571,667,711,791,807,841,869,889,931,979,
%U A064507 1071,1079,1107,1129,1189,1257,1273,1277,1297,1431,1437,1583,1611,1639
%N A064507 Numbers that yield primes when cast in their own base.
%e A064507 E.g. 21 in base 21 is 2*21+1=43, 111 is 1*111*111+1*111+1=12433, etc.
%Y A064507 Cf. A064508.
%K A064507 nonn
%O A064507 0,1
%A A064507 Jon Perry (perry@globalnet.co.uk), Oct 06 2001
%I A009727
%S A009727 1,1,21,27,6121,153559,13064637,996722035,114653539281,15657589156655,
%T A009727 2678537704263781,548432789294290763,133426766351448919481,
%U A009727 37955932970963263679495,12491453691407386059683341
%V A009727 1,-1,21,27,6121,153559,13064637,996722035,114653539281,15657589156655,
%W A009727 2678537704263781,548432789294290763,133426766351448919481,
%X A009727 37955932970963263679495,12491453691407386059683341
%N A009727 Expansion of tan(x).cos(sin(x)).
%t A009727 Tan[ x ]*Cos[ Sin[ x ] ] (* Odd Part *)
%K A009727 sign,done
%O A009727 0,3
%A A009727 R. H. Hardin (rhh@research.bell-labs.com)
%E A009727 Extended with signs 03/97 by Olivier Gerard.
%I A048012
%S A048012 0,0,0,0,0,0,0,0,0,21,28,36,144,180,220,550,660,780,2275,2821,3465,
%T A048012 13230,16940,21420,74256,94248,118104,367080,461700,574644,1824690,
%U A048012 2310000,2899380,9684840,12357532,15634300,51798175,66098175
%N A048012 Number of nonempty subsets of {1,2,...,n} in which exactly 3/5 of the elements are <= (n-1)/3.
%K A048012 nonn
%O A048012 1,10
%A A048012 Clark Kimberling, ck6@cedar.evansville.edu
%I A048067
%S A048067 0,0,0,0,0,0,0,0,0,21,28,112,144,360,450,1110,1430,4235,5775,17556,
%T A048067 24388,68328,94731,251160,349300,923230,1301080,3482710,4976070,
%U A048067 13287846,19136832,50391978,72898098
%N A048067 Number of nonempty subsets of {1,2,...,n} in which exactly 3/5 of the elements are <= (n-4)/2.
%K A048067 nonn
%O A048067 1,10
%A A048067 Clark Kimberling, ck6@cedar.evansville.edu
%I A004196
%S A004196 21,29,35,41,45,49,53,59,61,69,77,83,89,101,115,157
%N A004196 The numbers not expressible as the sum of 4 distinct nonzero squares can be written D*4^n union E. This is E.
%H A004196 Index entries for sequences related to sums of squares
%Y A004196 Cf. A004195, A004441.
%K A004196 nonn,fini,full
%O A004196 1,1
%A A004196 njas, Dan Hoey (Hoey@aic.nrl.navy.mil)
%I A027342
%S A027342 21,29,40,53,72,94,124,161,209,267,343,434,550,691,867,1079,1344,
%T A027342 1661,2051,2520,3091,3773,4602,5587,6774,8185,9874,11873,14259,
%U A027342 17072,20411,24343,28989,34440,40864,48378,57198,67497,79543
%N A027342 Partitions of n that do not contain 8 as a part.
%K A027342 nonn
%O A027342 8,1
%A A027342 Clark Kimberling, ck6@cedar.evansville.edu
%I A039372
%S A039372 21,29,102,110,165,174,189,190,193,194,195,196,197,201,210,219,228,237,
%T A039372 245,254,261,262,265,266,267,268,269,281,290,299,308,317,345,353,426,
%U A039372 434,507,515,588,596,669,677,750,758,831,839,894,903,918,919,922
%N A039372 Representation in base 9 has same nonzero number of 2's and 3's.
%K A039372 nonn,base,easy
%O A039372 0,1
%A A039372 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043195
%S A043195 21,29,102,110,183,189,261,272,345,353,426,434,507,515,588,596,
%T A043195 669,677,750,758,831,839,912,918,990,1001,1074,1082,1155,1163,
%U A043195 1236,1244,1317,1325,1398,1406,1479,1487,1560,1568,1641,1647
%N A043195 2 and 3 occur juxtaposed in the base 9 representation of n but not of n-1.
%K A043195 nonn,base
%O A043195 1,1
%A A043195 Clark Kimberling, ck6@cedar.evansville.edu
%I A043975
%S A043975 21,29,102,110,183,197,269,272,345,353,426,434,507,515,588,596,
%T A043975 669,677,750,758,831,839,912,926,998,1001,1074,1082,1155,1163,
%U A043975 1236,1244,1317,1325,1398,1406,1479,1487,1560,1568,1641,1655
%N A043975 2 and 3 occur juxtaposed in the base 9 representation of n but not of n+1.
%K A043975 nonn,base
%O A043975 1,1
%A A043975 Clark Kimberling, ck6@cedar.evansville.edu
%I A031889
%S A031889 21,31,43,49,93,99,105,111,141,169,201,211,267,273,303,327,367,391,
%T A031889 399,415,421,427,483,495,517,535,559,651,679,685,699,723,735,787,
%U A031889 873,903,1029,1093,1101,1123,1189,1209,1281,1291,1309,1329,1395
%N A031889 Lucky numbers with size of gaps equal to 6 (upper terms).
%Y A031889 Cf. A000959.
%K A031889 nonn
%O A031889 0,1
%A A031889 Patrick De Geest (pdg@worldofnumbers.com)
%I A034101
%S A034101 21,31,43,57,73,91,92,111,112,133,134,157,158,183,184,211,212,213,241,
%T A034101 242,243,273,274,275,307,308,309,343,344,345,381,382,383,384,421,422,
%U A034101 423,424,463,464,465,466,507,508,509,510,553,554,555,556,601,602,603
%N A034101 Decimal part of square root of a(n) starts with digit 5.
%Y A034101 Cf. A034111.
%K A034101 nonn,easy,base
%O A034101 0,1
%A A034101 Patrick De Geest (pdg@worldofnumbers.com), Sep 1998.
%I A034111
%S A034111 21,31,43,57,73,91,111,133,157,183,211,241,273,307,343,381,421,463,507,
%T A034111 553,601,651,703,757,813,871,931,993,1057,1123,1191,1261,1333,1407,
%U A034111 1483,1561,1641,1723,1807,1893,1981,2071,2163,2257,2353,2451,2551,2653
%N A034111 Decimal part of square root of a(n) starts with 5: first term of runs.
%C A034111 Agrees with sequence A002061 (n^2-n+1 = central polygonal numbers) starting with sixth term.
%Y A034111 Cf. A034101.
%K A034111 nonn,base
%O A034111 0,1
%A A034111 Patrick De Geest (pdg@worldofnumbers.com), Sep 1998.
%I A032585
%S A032585 1,21,31,51,111,141,151,171,201,211,231,241,261,321,331,361,391,421,
%T A032585 451,511,541,591,601,621,631,651,741,781,801,831,841,931,961,981,991,
%U A032585 1011,1021,1041,1101,1201,1231,1251,1261,1281,1291,1401,1441,1471,1491
%N A032585 Lucky numbers ending with digit 1.
%Y A032585 Cf. A000959.
%K A032585 nonn,base
%O A032585 0,2
%A A032585 Patrick De Geest (pdg@worldofnumbers.com), april 1998.
%I A032013
%S A032013 0,1,1,1,21,31,113,169,8053,15871,71325,300147,816401,63105953,
%T A032013 161203747,856049593,4050514725,25570388671,80377109117,
%U A032013 12126315199099,36747628912981,233849676829957,1239662165799711
%N A032013 Partition n labeled elements into sets of different sizes of at least 2 and order the sets.
%H A032013 C. G. Bower, Transforms (2)
%F A032013 "AGJ" (ordered, elements, labeled) transform of 0,1,1,1...
%K A032013 nonn
%O A032013 1,5
%A A032013 Christian G. Bower (bowerc@usa.net)
%I A019423
%S A019423 1,21,31,651,889,3210,3498,3710,3882,3910,4310,4922,4982,5182,5457,5885,
%T A019423 6035,6095,6307,6797,7117,7327,7597,24573,27559,71193,82110,90510,94981,
%U A019423 97410,98671,99301,99510,100110,103362,104622,107778,108438,108822
%N A019423 Sum of divisors of n is a fifth power.
%H A019423 Index entries for continued fractions for constants
%K A019423 nonn
%O A019423 1,2
%A A019423 dww
%I A002639 M5108 N2212
%S A002639 1,1,21,31,6257,10293,279025,483127,435506703,776957575,22417045555,
%T A002639 40784671953
%N A002639 Coefficients of Jacobi nome.
%D A002639 Guide to Tables, Math. Tables Other Aids Computation, 3 (1948), Section III, p. 234.
%D A002639 C. Hermite, Oeuvres. Vol. 4, Gauthier-Villars, Paris, 1917, p. 477.
%H A002639 J. Tannery and J. Molk, El\'{e}ments de la Th\'{e}orie des Fonctions Elliptiques (Vol 4), Gauthier-Villars, Paris, 1902, p. 141.
%Y A002639 Cf. A002103 (where there are further references).
%K A002639 nonn,more,nice
%O A002639 1,3
%A A002639 njas
%I A054282
%S A054282 21,32,33,47,57,58,60,62,70,71,96,97,105,106,130,131,150,155,167,177,
%T A054282 234,243,246,248,249,255,261,300,309,314,316,318,343,345,350,352,386,
%U A054282 388,401,406,419,423,445,449,451
%N A054282 Positions of 6's in the decimal expansion of exp(1).
%K A054282 nonn
%O A054282 0,1
%A A054282 Simon Plouffe (plouffe@math.uqam.ca), Feb 20, 2000
%I A035137
%S A035137 21,32,43,54,65,76,87,98,201,1031,1041,1042,1051,1052,1053,1061,1062,
%T A035137 1063,1064,1071,1072,1073,1074,1075,1081,1082,1083,1084,1085,1086,1091,
%U A035137 1092,1093,1094,1095,1096,1097,1099,1101,1103,1104,1105,1106,1107,1108
%N A035137 Not the sum of 2 palindromes.
%H A035137 P. De Geest, World!Of Numbers
%Y A035137 Cf. A014091, A014092.
%K A035137 nonn,base
%O A035137 0,1
%A A035137 Patrick De Geest (pdg@worldofnumbers.com), Nov 1998.
%I A036382
%S A036382 21,33,35,39,65,69,75,77,87,91,93,105,129,133,135,141,143,145,147,155,
%T A036382 159,161,165,175,177,183,189,195,203,217,259,261,265,267,273,275,279,
%U A036382 285,287,291,295,297,299,301,303,305,309,315,319,321,325,327,329,339
%N A036382 Odd split numbers: have a non-trivial factorization n=ab, lcm[ a,b ]=1, so that g(a)+g(b)<=g(n), where g(x) is binary order of x.
%C A036382 All even numbers are split numbers, except that prime powers - here powers of 2 - are by definition excluded.
%e A036382 s=39 is a split number since s=39=3*13, LCM[ 3,13 ]=1 and g(3)+g(13)=2+4=g(39).
%Y A036382 Cf. A029827, A036377-A036390.
%K A036382 nonn
%O A036382 1,1
%A A036382 Labos E. (labos@ana1.sote.hu)
%I A026068
%S A026068 21,33,49,68,90,116,145,179,217,259,306,357,414,476,543,616,694,779,870,967,1071,
%T A026068 1181,1299,1424,1556,1696,1843,1999,2163,2335,2516,2705,2904,3112,3329,3556,3792,
%U A026068 4039,4296,4563,4841,5129,5740,6062,6396,6741,7099,7469,7851,8246,8653,9074,9508
%N A026068 (d(n)-r(n))/5, where d = A026066 and r is the periodic sequence with fund. period (0,3,1,0,1).
%K A026068 nonn
%O A026068 7,1
%A A026068 Clark Kimberling (ck6@cedar.evansville.edu)
%I A016105
%S A016105 21,33,57,69,77,93,129,133,141,161,177,201,209,213,217,237,249,253,301,
%T A016105 309,321,329,341,381,393,413,417,437,453,469,473,489,497,501,517,537,
%U A016105 553,573,581,589,597,633,649,669,681,713,717,721,737,749,753,781,789
%N A016105 Blum numbers: of form P*Q where P&Q are distinct primes congruent to 3 (mod 4).
%K A016105 nonn,easy,nice
%O A016105 1,1
%A A016105 Robert G. Wilson v (rgwv@kspaint.com)
%E A016105 More terms from Erich Friedman (erich.friedman@stetson.edu).
%I A032603
%S A032603 21,33,57,79,1113,1315,1721,1925,2331,2933,3137,3743,4149,4351,4763,
%T A032603 5367,5969,6173,6775,7179,7387,7993,8399,89105,97111,101115,103127,
%U A032603 107129,109133,113135,127141,131151,137159,139163,149169,151171,157189
%N A032603 Concatenation of n-th prime number and n-th lucky number.
%Y A032603 Cf. A032604, A000959.
%K A032603 nonn
%O A032603 0,1
%A A032603 Patrick De Geest (pdg@worldofnumbers.com), may 1998.
%I A033901
%S A033901 21,33,66,132,255,510,525,780,858,1446,2892,5181,6339,9708,10497,11976,
%T A033901 23655,47211,58458,104046,105492,117951,229530,251889,377778,755556,
%U A033901 1311123,2422356,4645812,5890380,5926269,8182968,9451857,10907646
%N A033901 Sort then Add!.
%Y A033901 Cf. A033860.
%K A033901 nonn,base,easy
%O A033901 1,1
%A A033901 njas, dww
%I A033656
%S A033656 21,33,66,132,363,726,1353,4884,9768,18447,92928,175857,
%T A033656 934428,1758867,9447438,17794887,96644658,182289327,906271608,
%U A033656 1712444217,8836886388,17673772776,85401510447,159803020905
%N A033656 Reverse and Add!.
%K A033656 nonn
%O A033656 1,1
%A A033656 njas
%I A032792
%S A032792 1,21,33,133,261,385,645,1281,1365,2133,2821,4033,4485,7701,11781,
%T A032792 16725,19521,20833,21505,22533,22881,27265,29205,32865,36741,37185,
%U A032792 43681,47125,48133,50181,64533,65121,69921,73633,75525,79381,88065
%N A032792 Quotients of n(n+1)(n+2) / n+(n+1)+(n+2) are lucky numbers.
%Y A032792 Cf. A032791.
%K A032792 nonn
%O A032792 0,2
%A A032792 Patrick De Geest (pdg@worldofnumbers.com), may 1998.
%I A053409
%S A053409 21,34,55,377,4181,17711,121393,1346269,5702887,165580141,53316291173,
%T A053409 956722026041,2504730781961,308061521170129,806515533049393,
%U A053409 14472334024676221,1779979416004714189,19740274219868223167
%N A053409 Semi-prime Fibonacci numbers.
%e A053409 a(1)=21 because 21 is the product of two primes.
%Y A053409 Cf. A000045, A000040, A001358.
%K A053409 nonn
%O A053409 1,1
%A A053409 G. L. Honaker, Jr. (curios@bvub.com), Jan 09 2000
%I A008946
%S A008946 1,21,35,35,90,140,189,210,280,280,280,280,315,315,420,
%T A008946 560,640,640,729,896
%N A008946 Degrees of irreducible representations of group U4(3).
%D A008946 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%o A008946 (GAP) Display(CharacterTable("U4(3)"));
%K A008946 nonn,fini,full
%O A008946 0,2
%A A008946 njas
%I A001491
%S A001491 21,36,55,60,67,68,92,93,125
%N A001491 Opus numbers of Beethoven's nine symphonies.
%C A001491 Called the "Trice sequence" in C. A. Pickover's "Mazes of the Mind", 1992, p. 351.
%H A001491 Beethoven web site
%K A001491 fini,full,nonn
%O A001491 1,1
%A A001491 njas, GEIER@LaDune.Westfalen.De (Michael Gierhake)
%E A001491 This was a mystery for many years - even Clifford Pickover could not recall the explanation. It was finally identified in Oct 1998 by Derek Holt (dfh@maths.warwick.ac.uk)
%I A043683
%S A043683 21,37,41,42,43,45,53,69,73,74,75,77,81,82,83,84,86,87,89,90,
%T A043683 91,93,101,105,106,107,109,117,133,137,138,139,141,145,146,147,
%U A043683 148,150,151,153,154,155,157,161,162,163,164,166,167,168,172
%N A043683 (1/2)(n-th number whose base 2 representation has exactly 6 runs).
%K A043683 nonn,base
%O A043683 1,1
%A A043683 Clark Kimberling, ck6@cedar.evansville.edu
%I A043572
%S A043572 21,37,41,43,45,53,69,73,75,77,81,83,87,89,91,93,101,105,107,
%T A043572 109,117,133,137,139,141,145,147,151,153,155,157,161,163,167,
%U A043572 175,177,179,183,185,187,189,197,201,203,205,209,211,215,217
%N A043572 Base 2 representation has exactly 5 runs.
%K A043572 nonn,base
%O A043572 1,1
%A A043572 Clark Kimberling, ck6@cedar.evansville.edu
%I A043728
%S A043728 21,37,41,43,45,53,69,73,75,77,81,83,87,89,91,93,101,105,107,
%T A043728 109,117,133,137,139,141,145,147,151,153,155,157,161,163,167,
%U A043728 175,177,179,183,185,187,189,197,201,203,205,209,211,215,217
%N A043728 Number of runs in the base 2 representation of n is congruent to 0 mod 5.
%K A043728 nonn,base
%O A043728 1,1
%A A043728 Clark Kimberling, ck6@cedar.evansville.edu
%I A043738
%S A043738 21,37,41,43,45,53,69,73,75,77,81,83,87,89,91,93,101,105,107,
%T A043738 109,117,133,137,139,141,145,147,151,153,155,157,161,163,167,
%U A043738 175,177,179,183,185,187,189,197,201,203,205,209,211,215,217
%N A043738 Number of runs in the base 2 representation of n is congruent to 5 mod 6.
%K A043738 nonn,base
%O A043738 1,1
%A A043738 Clark Kimberling, ck6@cedar.evansville.edu
%I A043744
%S A043744 21,37,41,43,45,53,69,73,75,77,81,83,87,89,91,93,101,105,107,
%T A043744 109,117,133,137,139,141,145,147,151,153,155,157,161,163,167,
%U A043744 175,177,179,183,185,187,189,197,201,203,205,209,211,215,217
%N A043744 Number of runs in the base 2 representation of n is congruent to 5 mod 7.
%K A043744 nonn,base
%O A043744 1,1
%A A043744 Clark Kimberling, ck6@cedar.evansville.edu
%I A043751
%S A043751 21,37,41,43,45,53,69,73,75,77,81,83,87,89,91,93,101,105,107,
%T A043751 109,117,133,137,139,141,145,147,151,153,155,157,161,163,167,
%U A043751 175,177,179,183,185,187,189,197,201,203,205,209,211,215,217
%N A043751 Number of runs in the base 2 representation of n is congruent to 5 mod 8.
%K A043751 nonn,base
%O A043751 1,1
%A A043751 Clark Kimberling, ck6@cedar.evansville.edu
%I A043759
%S A043759 21,37,41,43,45,53,69,73,75,77,81,83,87,89,91,93,101,105,107,
%T A043759 109,117,133,137,139,141,145,147,151,153,155,157,161,163,167,
%U A043759 175,177,179,183,185,187,189,197,201,203,205,209,211,215,217
%N A043759 Number of runs in the base 2 representation of n is congruent to 5 mod 9.
%K A043759 nonn,base
%O A043759 1,1
%A A043759 Clark Kimberling, ck6@cedar.evansville.edu
%I A043768
%S A043768 21,37,41,43,45,53,69,73,75,77,81,83,87,89,91,93,101,105,107,
%T A043768 109,117,133,137,139,141,145,147,151,153,155,157,161,163,167,
%U A043768 175,177,179,183,185,187,189,197,201,203,205,209,211,215,217
%N A043768 Number of runs in the base 2 representation of n is congruent to 5 mod 10.
%K A043768 nonn,base
%O A043768 1,1
%A A043768 Clark Kimberling, ck6@cedar.evansville.edu
%I A050782
%S A050782 0,1,1,1,1,1,1,1,1,1,0,1,21,38,18,35,17,16,14,9,0,12,1,7,29,21,19,37,9,
%T A050782 8,0,14,66,1,8,15,7,3,13,15,0,16,6,23,1,13,9,3,44,7,0,19,13,4,518,1,11,
%U A050782 3,4,13,0,442,7,4,33,9,1,11,4,6,0,845,88,4,3,7,287,1,11,6,0,12345679,8
%N A050782 Smallest multiplier m (>0) such that mn is palindromic (or zero if no such m exists).
%C A050782 Multiples of 81 require the largest multipliers.
%H A050782 P. De Geest, World!Of Numbers
%e A050782 E.g. a(81) -> 81 * 12345679 = 999999999 and a palindrome.
%Y A050782 Cf. A002113, A050810.
%K A050782 nonn,base,nice
%O A050782 0,13
%A A050782 Patrick De Geest (pdg@worldofnumbers.com), Oct 1999.
%I A061906
%S A061906 1,1,1,1,1,1,1,1,1,1,1,1,21,38,18,35,17,16,14,9,1,12,1,7,29,21,19,37,9,
%T A061906 8,1,14,66,1,8,15,7,3,13,15,1,16,6,23,1,13,9,3,44,7,1,19,13,4,518,1,11,
%U A061906 3,4,13,1,442,7,4,33,9,1,11,4,6,1,845,88,4,3,7,287,1,11,6,1,12345679,8
%N A061906 Obtain m by omitting trailing zeros from n; a(n) = smallest k such that k*m is a palindrome.
%C A061906 Every positive integer is a factor of a palindrome, unless it is a multiple of 10 (D. G. Radcliffe, see Links).
%C A061906 Every integer n has a multiple of the form 99...9900...00. To see that n has a multiple that's a palindrome (allowing 0's on the left) with even digits, let 9n divide 99...9900...00; then n divides 22...2200...00. - Dean Hickerson, Jun 29, 2001.
%H A061906 P. De Geest, Smallest multipliers to make a number palindromic.
%e A061906 For n = 30 we have m = 3, 1*m = 3 is a palindrome, so a(30) = 1. For n = m = 12 the smallest palindromic multiple is 21*m = 252, so a(12) = 21.
%o A061906 (ARIBAS): stop := 20000000; for n := 0 to maxarg do k:= 1; test := true; while test and k < stop do mp := omit_trailzeros(n)*k; if test := mp <> int_reverse(mp) then inc(k); end; end; if k < stop then write(k," "); else write(-1," "); end; end;
%Y A061906 Cf. A050782, A062293, A061915, A061916, A061816. Values of k*m are given in A061906.
%K A061906 base,easy,nonn
%O A061906 0,13
%A A061906 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jun 25 2001
%I A020220
%S A020220 21,39,65,91,93,105,217,231,273,301,341,403,451,465,559,561,651,861,1001,
%T A020220 1085,1105,1209,1271,1333,1365,1393,1661,1729,2465,2587,2701,2821,3171,
%U A020220 3731,3781,3913,4033,4123,4371,4641,4681,5565,6045,6169,6191,6697,7161
%N A020220 Pseudoprimes to base 92.
%H A020220 Index entries for sequences related to pseudoprimes
%K A020220 nonn
%O A020220 1,1
%A A020220 dww
%I A067344
%S A067344 1,21,41,120,242,312,323,401,501,1040,1114,1141,1204,1214,1233,1241,
%T A067344 1304,1503,2033,2115,2133,2140,2403,3010,3014,3124,3211,3304,3322,4001,
%U A067344 4012,4121,4301,4310,5130,10044,10214,10242,10320,10324,11042,11115
%N A067344 Sum of decimal digits of square of divisors of n equals sum of square of digits of n.
%F A067344 A007953[A001157(n)]=A007953[n]^2
%e A067344 n=51223, SquareSumDigit=25+1+4+4+9=43, Sigma[2,51223]=2623908580 with digitsum=43.
%t A067344 Do[s=Apply[Plus,IntegerDigits[DivisorSigma[2,n]]]- Apply[Plus,IntegerDigits[n]^2]; If[Equal[s,0],Print[n]],{n,1,10000}]
%Y A067344 Cf. A067342, A067343, A007953, A001157.
%K A067344 base,nonn
%O A067344 1,2
%A A067344 Labos E. (labos@ana1.sote.hu), Jan 16 2002
%I A053428
%S A053428 1,1,21,41,461,1281,10501,36121,246141,968561,5891381,25262601,
%T A053428 143090221,648342241,3510146661,16476991481,86679924701,416219754321,
%U A053428 2149818248341,10474213334761,53470578301581,262954844996801
%N A053428 A second order recurrence.
%C A053428 a(n)=a(n-1)+20a(n-2), n>1; a(0)=1, a(1)=1.
%D A053428 A. H. Beiler, Recreations in the Theory of Numbers,Dover,N.Y.,1964,pps.194-196.
%D A053428 A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly,Vol.15,No.1,1979,pp.21,24,29.
%F A053428 a(n)=[(5^(n+1))-(-4)^(n+1)]/9.
%Y A053428 Cf. A001045, A015441, A053404.
%K A053428 easy,nonn
%O A053428 0,3
%A A053428 Barry E. Williams, Jan 10 2000
%E A053428 More terms from James A. Sellers (sellersj@math.psu.edu), Feb 02 2000
%I A040420
%S A040420 21,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,
%T A040420 42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,
%U A040420 42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42
%N A040420 Continued fraction for sqrt(442).
%H A040420 Index entries for continued fractions for constants
%p A040420 with(numtheory): Digits:=300: convert(evalf(sqrt(442)),confrac);
%K A040420 nonn,cofr,easy
%O A040420 0,1
%A A040420 njas
%I A037978
%S A037978 21,42,63,64,84,86,87,106,127,128,149,168,169,171,191,192,213,
%T A037978 234,252,253,254,257,258,259,277,298,319,320,336,337,338,339,
%U A037978 344,345,346,347,348,349,350,351,362,383,384,405,424,425,427
%N A037978 n-th number whose maximal base 4 run length is 3.
%K A037978 nonn,base
%O A037978 1,1
%A A037978 Clark Kimberling, ck6@cedar.evansville.edu
%I A008603
%S A008603 0,21,42,63,84,105,126,147,168,189,210,231,252,273,294,
%T A008603 315,336,357,378,399,420,441,462,483,504,525,546,567,
%U A008603 588,609,630,651,672,693,714,735,756,777,798,819,840
%N A008603 Multiples of 21.
%H A008603 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 333
%K A008603 nonn
%O A008603 0,2
%A A008603 njas
%I A001682 M5109 N2213
%S A001682 21,42,65,86,109,130,151,174,195,218,239,262,283,304,327,348,371,
%T A001682 392,415,436,457,480,501,524,545,568,589,610,633,654,677,698,721,742,763
%N A001682 3^n, 3^(n+1) and 3^(n+2) have same number of digits.
%D A001682 Problem E1238, Amer. Math. Monthly, 64 (1957), 367.
%K A001682 nonn,base,easy,nice
%O A001682 1,1
%A A001682 njas
%I A039344
%S A039344 21,42,85,106,133,141,157,165,168,169,171,172,174,175,181,189,213,234,
%T A039344 277,298,322,330,336,337,339,340,342,343,346,354,370,378,405,426,469,
%U A039344 490,533,554,597,618,645,653,669,677,680,681,683,684,686,687,693,701
%N A039344 Representation in base 8 has same nonzero number of 2's and 5's.
%K A039344 nonn,base,easy
%O A039344 0,1
%A A039344 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043167
%S A043167 21,42,85,106,149,168,213,234,277,298,336,362,405,426,469,490,
%T A043167 533,554,597,618,661,680,725,746,789,810,848,874,917,938,981,
%U A043167 1002,1045,1066,1109,1130,1173,1192,1237,1258,1301,1322,1344
%N A043167 2 and 5 occur juxtaposed in the base 8 representation of n but not of n-1.
%K A043167 nonn,base
%O A043167 1,1
%A A043167 Clark Kimberling, ck6@cedar.evansville.edu
%I A043947
%S A043947 21,42,85,106,149,175,213,234,277,298,343,362,405,426,469,490,
%T A043947 533,554,597,618,661,687,725,746,789,810,855,874,917,938,981,
%U A043947 1002,1045,1066,1109,1130,1173,1199,1237,1258,1301,1322,1407
%N A043947 2 and 5 occur juxtaposed in the base 8 representation of n but not of n+1.
%K A043947 nonn,base
%O A043947 1,1
%A A043947 Clark Kimberling, ck6@cedar.evansville.edu
%I A041870
%S A041870 21,43,64,107,171,962,2095,28197,30292,119073,149365,716533,
%T A041870 865898,1582431,7195622,8778053,33529781,42307834,583531623,
%U A041870 1209371080,6630387023,7839758103,14470145126,22309903229
%N A041870 Numerators of continued fraction convergents to sqrt(457).
%Y A041870 Cf. A041871.
%K A041870 nonn,cofr,easy
%O A041870 0,1
%A A041870 njas
%I A041868
%S A041868 21,43,64,299,363,1025,43413,87851,131264,612907,744171,
%T A041868 2101249,88996629,180094507,269091136,1256459051,1525550187,
%U A041868 4307559425,182443046037,369193651499,551636697536,2575740441643
%N A041868 Numerators of continued fraction convergents to sqrt(456).
%Y A041868 Cf. A041869.
%K A041868 nonn,cofr,easy
%O A041868 0,1
%A A041868 njas
%I A041874
%S A041874 21,43,107,150,707,14997,60695,75692,212079,499850,21205779,
%T A041874 42911408,107028595,149940003,706788607,14992500750,60676791607,
%U A041874 75669292357,212015376321,499700044999,21199417266279,42898534577557
%N A041874 Numerators of continued fraction convergents to sqrt(459).
%Y A041874 Cf. A041875.
%K A041874 nonn,cofr,easy
%O A041874 0,1
%A A041874 njas
%I A041872
%S A041872 21,43,107,4537,9181,22899,970939,1964777,4900493,207785483,
%T A041872 420471459,1048728401,44467064301,89982857003,224432778307,
%U A041872 9516159545897,19256751870101,48029663286099,2036502609886259
%N A041872 Numerators of continued fraction convergents to sqrt(458).
%Y A041872 Cf. A041873.
%K A041872 nonn,cofr,easy
%O A041872 0,1
%A A041872 njas
%I A041876
%S A041876 21,43,193,622,815,2252,23335,48922,72257,265693,1135029,
%T A041876 2535751,107636571,217808893,978872143,3154425322,4133297465,
%U A041876 11421020252,118343499985,248108020222,366451520207,1347462580843
%N A041876 Numerators of continued fraction convergents to sqrt(460).
%Y A041876 Cf. A041877.
%K A041876 nonn,cofr,easy
%O A041876 0,1
%A A041876 njas
%I A041878
%S A041878 21,43,365,3693,4058,7751,11809,19560,31369,50929,82298,
%T A041878 133227,1414568,11449771,24314110,1032642391,2089598892,
%U A041878 17749433527,179583934162,197333367689,376917301851,574250669540
%N A041878 Numerators of continued fraction convergents to sqrt(461).
%Y A041878 Cf. A041879.
%K A041878 nonn,cofr,easy
%O A041878 0,1
%A A041878 njas
%I A041880
%S A041880 21,43,1827,3697,157101,317899,13508859,27335617,1161604773,
%T A041880 2350545163,99884501619,202119548401,8588905534461,17379930617323,
%U A041880 738545991462027,1494471913541377,63506366360199861,128507204633941099
%N A041880 Numerators of continued fraction convergents to sqrt(462).
%Y A041880 Cf. A041881.
%K A041880 nonn,cofr,easy
%O A041880 0,1
%A A041880 njas
%I A063319
%S A063319 1,21,44,66,88,110,132,152,176,196,220,242,264,282,308,328,352,
%T A063319 372,396,414,440,458,484,504,528,544,572,590,616,634,660,676,704,
%U A063319 720,748,766,792,806,836,852,880,896,924,938,968,982,1012,1028
%V A063319 -1,21,44,66,88,110,132,152,176,196,220,242,264,282,308,328,352,
%W A063319 372,396,414,440,458,484,504,528,544,572,590,616,634,660,676,704,
%X A063319 720,748,766,792,806,836,852,880,896,924,938,968,982,1012,1028
%N A063319 Dimension of the space of weight n cuspidal newforms for Gamma_1( 46 ).
%H A063319 William A. Stein (was@math.berkeley.edu), Dimensions of the spaces S_k^{new}(Gamma_1(N))
%H A063319 William A. Stein (was@math.berkeley.edu), The modular forms database
%K A063319 sign,done
%O A063319 2,2
%A A063319 njas, Jul 14 2001
%I A003857
%S A003857 1,21,45,45,55,99,154,210,231,280,280,385
%N A003857 Degrees of irreducible representations of group M22.
%D A003857 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%o A003857 (GAP) Display(CharacterTable("M22"));
%K A003857 nonn,fini,full
%O A003857 1,2
%A A003857 njas
%I A063500
%S A063500 21,45,63,95,224,135,301,95,189,161,260,115,400,165,476,195,620,145,
%T A063500 644,203,640,285,343,155,728,185,567,155,560,301,860,185,1000,329,1892,
%U A063500 235
%N A063500 a(n) is the least composite solution of Phi[x+6n]=Phi[x]+6n.
%F A063500 Smallest values satisfying A000010[a(n)+6n]=A000010[a(n)]+6n relation.
%e A063500 n=100, d=600=6n, a(100)=671=11.61, Phi[671]=600, Phi[671+600]=Phi[1271]=(31-1).(41-1)=600+600=Phi[671]+d.
%Y A063500 A000010, A054904, A054905.
%K A063500 more,nonn
%O A063500 1,1
%A A063500 Labos E. (labos@ana1.sote.hu), Jul 30 2001
%E A063500 Next term (a(37)) exceeds 10^5 - Matthew M. Conroy (doctormatt@earthlink.net), Sep 13 2001
%I A044098
%S A044098 21,46,71,96,105,121,146,171,196,221,230,246,271,296,321,346,
%T A044098 355,371,396,421,446,471,480,496,521,525,571,596,605,621,646,
%U A044098 671,696,721,730,746,771,796,821,846,855,871,896,921,946,971
%N A044098 String 4,1 occurs in the base 5 representation of n but not of n-1.
%K A044098 nonn,base
%O A044098 1,1
%A A044098 Clark Kimberling, ck6@cedar.evansville.edu
%I A044479
%S A044479 21,46,71,96,109,121,146,171,196,221,234,246,271,296,321,346,359,371,
%T A044479 396,421,446,471,484,496,521,549,571,596,609,621,646,671,696,721,734,
%U A044479 746,771,796,821,846,859,871,896,921,946,971
%N A044479 String 4,1 occurs in the base 5 representation of n but not of n+1.
%K A044479 nonn,base
%O A044479 1,1
%A A044479 Clark Kimberling, ck6@cedar.evansville.edu
%I A020178
%S A020178 21,49,51,119,133,147,231,301,357,561,637,697,793,817,833,861,931,1037,
%T A020178 1281,1649,1729,2009,2041,2047,2107,2499,2501,2701,2821,2989,3201,3281,
%U A020178 3913,3977,4753,5461,5719,6601,7693,7701,8041,8113,8911,9061,9073,9331
%N A020178 Pseudoprimes to base 50.
%H A020178 Index entries for sequences related to pseudoprimes
%K A020178 nonn
%O A020178 1,1
%A A020178 dww
%I A053178
%S A053178 1,21,51,81,91,111,121,141,161,171,201,221,231,261,291,301,321,341,351,
%T A053178 361,371,381,391,411,441,451,471,481,501,511,531,551,561,581,591,611,
%U A053178 621,651,671,681,711,721,731,741,771,781,791,801,831,841,851,861,871
%N A053178 Numbers ending in 1 which are not prime.
%e A053178 a(4)=91 may look prime to some, but is composite.
%K A053178 easy,nonn,base
%O A053178 1,2
%A A053178 Enoch Haga (EnochHaga@msn.com), Feb 29 2000
%I A043133
%S A043133 21,52,70,101,119,147,168,199,217,248,266,297,315,346,364,395,
%T A043133 413,444,462,490,511,542,560,591,609,640,658,689,707,738,756,
%U A043133 787,805,833,854,885,903,934,952,983,1001,1029,1081,1099,1130
%N A043133 0 and 3 occur juxtaposed in the base 7 representation of n but not of n-1.
%K A043133 nonn,base
%O A043133 1,1
%A A043133 Clark Kimberling, ck6@cedar.evansville.edu
%I A039310
%S A039310 21,52,70,101,119,148,149,151,152,153,154,161,175,182,189,199,217,248,
%T A039310 266,297,315,353,360,365,366,368,369,370,374,381,388,395,413,444,462,
%U A039310 491,492,494,495,496,497,504,518,525,532,542,560,591,609,640,658,696
%N A039310 Representation in base 7 has same nonzero number of 0's and 3's.
%K A039310 nonn,base,easy
%O A039310 0,1
%A A039310 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043913
%S A043913 21,52,70,101,119,153,168,199,217,248,266,297,315,346,370,395,
%T A043913 413,444,462,496,511,542,560,591,609,640,658,689,713,738,756,
%U A043913 787,805,839,854,885,903,934,952,983,1001,1077,1081,1099,1130
%N A043913 0 and 3 occur juxtaposed in the base 7 representation of n but not of n+1.
%K A043913 nonn,base
%O A043913 1,1
%A A043913 Clark Kimberling, ck6@cedar.evansville.edu
%I A007796
%S A007796 21,53,117,1713,2319,3129,4137,4743,5953,6761,7371,8379,
%T A007796 9789,103101,109107,127113,137131,149139,157151,167163,179173,
%U A007796 191181,197193,211199,227223
%N A007796 Pairs of primes in reverse order, starting at 1.
%K A007796 nonn,base
%O A007796 1,1
%A A007796 bmoore@artemis.ess.ucla.edu (William B. Moore)
%I A043382
%S A043382 21,57,93,111,117,123,126,127,128,130,131,135,141,165,201,237,
%T A043382 273,309,327,333,339,342,343,344,346,347,351,357,381,417,453,
%U A043382 489,525,543,549,555,558,559,560,562,563,567,573,597,633,651
%N A043382 Number of 3's in base 6 is 2.
%K A043382 nonn,base
%O A043382 1,1
%A A043382 Clark Kimberling, ck6@cedar.evansville.edu
%I A044123
%S A044123 21,57,93,126,165,201,237,273,309,342,381,417,453,489,525,558,
%T A044123 597,633,669,705,741,756,813,849,885,921,957,990,1029,1065,1101,
%U A044123 1137,1173,1206,1245,1281,1317,1353,1389,1422,1461,1497,1533
%N A044123 String 3,3 occurs in the base 6 representation of n but not of n-1.
%K A044123 nonn,base
%O A044123 1,1
%A A044123 Clark Kimberling, ck6@cedar.evansville.edu
%I A044504
%S A044504 21,57,93,131,165,201,237,273,309,347,381,417,453,489,525,563,597,633,
%T A044504 669,705,741,791,813,849,885,921,957,995,1029,1065,1101,1137,1173,1211,
%U A044504 1245,1281,1317,1353,1389,1427,1461,1497,1533
%N A044504 String 3,3 occurs in the base 6 representation of n but not of n+1.
%K A044504 nonn,base
%O A044504 1,1
%A A044504 Clark Kimberling, ck6@cedar.evansville.edu
%I A020148
%S A020148 21,57,133,231,399,561,671,861,889,1281,1653,1729,1891,2059,2413,2501,
%T A020148 2761,2821,2947,3059,3201,4047,5271,5461,5473,5713,5833,6601,6817,7999,
%U A020148 8421,8911,11229,11557,11837,12801,13051,13981,14091,15251,15311,15841
%N A020148 Pseudoprimes to base 20.
%H A020148 Index entries for sequences related to pseudoprimes
%K A020148 nonn
%O A020148 1,1
%A A020148 dww
%I A037305
%S A037305 1,21,58,59,106,107,126,127,154,155,203,246,247,252,253,366,367,
%T A037305 378,379,394,395,414,415,444,445,462,463,492,493,498,499,504,
%U A037305 505,539,637,688,689,707,736,737,756,757,798,799,847,882,883
%N A037305 (sum of base 2 digits of n)=(sum of base 7 digits of n).
%K A037305 nonn,base
%O A037305 1,2
%A A037305 Clark Kimberling, ck6@cedar.evansville.edu
%I A051873
%S A051873 0,1,21,60,118,195,291,406,540,693,865,1056,1266,1495,1743,2010,
%T A051873 2296,2601,2925,3268,3630,4011,4411,4830,5268,5725,6201,6696,7210,
%U A051873 7743,8295,8866,9456,10065,10693,11340,12006,12691,13395,14118
%N A051873 21-gonal numbers.
%D A051873 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
%F A051873 a(n)=n(19n-17)/2.
%K A051873 nonn
%O A051873 0,3
%A A051873 njas, Dec 15 1999
%I A058100
%S A058100 1,0,21,62,162,378,819,1680,3276,6138,11145,19662,33840,57048,94362,
%T A058100 153432,245757,388218,605466,933414,1423614,2149586,3215844,4769544,
%U A058100 7016572
%N A058100 McKay-Thompson series of class 10D for Monster.
%D A058100 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058100 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058100 nonn
%O A058100 -1,3
%A A058100 njas, Nov 27 2000
%I A025525
%S A025525 21,63,104,84,102,82,76,112,94,122,88,108,110,138,148,172,111,130,156,
%T A025525 192,166,168,188,160,122,186,208,212,189,202,216,252,159,194,228,240,230,
%U A025525 216,292,308,222,250,316,324,262,266,332,364,239,282,316,384,294,274,364
%N A025525 Largest number that is not the sum of distinct numbers of form k^2 + n.
%K A025525 nonn
%O A025525 1,1
%A A025525 dww
%I A033850
%S A033850 21,63,147,189,441,567,1029,1323,1701,3087,3969,5103,7203,9261,11907,
%T A033850 15309,21609,27783,35721,45927,50421,64827,83349,107163,137781,151263,
%U A033850 194481,250047,321489,352947,413343,453789,583443,750141,964467
%N A033850 Prime factors are 3 and 7.
%K A033850 nonn
%O A033850 0,1
%A A033850 Jeff Burch (gburch@erols.com)
%I A033481
%S A033481 21,64,32,16,8,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,
%T A033481 4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,
%U A033481 4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1
%N A033481 3x+1 sequence beginning at 21.
%H A033481 Index entries for sequences related to 3x+1 (or Collatz) problem
%K A033481 nonn
%O A033481 0,1
%A A033481 Jeff Burch (jmburch@osprey.smcm.edu)
%I A041862
%S A041862 21,64,85,149,1575,22199,223565,245764,469329,1653751,69926871,
%T A041862 211434364,281361235,492795599,5209317225,73423236749,739441684715,
%U A041862 812864921464,1552306606179,5469784740001,231283265686221
%N A041862 Numerators of continued fraction convergents to sqrt(453).
%Y A041862 Cf. A041863.
%K A041862 nonn,cofr,easy
%O A041862 0,1
%A A041862 njas
%I A041860
%S A041860 21,64,85,489,1552,16009,49579,263904,313483,1204353,50896309,
%T A041860 153893280,204789589,1177841225,3738313264,38560973865,
%U A041860 119421234859,635667148160,755088383019,2900932297217,122594244866133
%N A041860 Numerators of continued fraction convergents to sqrt(452).
%Y A041860 Cf. A041861.
%K A041860 nonn,cofr,easy
%O A041860 0,1
%A A041860 njas
%I A041864
%S A041864 21,64,213,277,3814,7905,27529,62963,90492,153455,704312,
%T A041864 4379327,5083639,39964800,45048439,85013239,130061678,2686246799,
%U A041864 2816308477,5502555276,8318863753,63734601547,72053465300
%N A041864 Numerators of continued fraction convergents to sqrt(454).
%Y A041864 Cf. A041865.
%K A041864 nonn,cofr,easy
%O A041864 0,1
%A A041864 njas
%I A041866
%S A041866 21,64,2709,8191,346731,1048384,44378859,134184961,5680147221,
%T A041866 17174626624,727014465429,2198218022911,93052171427691,
%U A041866 281354732305984,11909950928279019,36011207517143041,1524380666648286741
%N A041866 Numerators of continued fraction convergents to sqrt(455).
%Y A041866 Cf. A041867.
%K A041866 nonn,cofr,easy
%O A041866 0,1
%A A041866 njas
%I A020211
%S A020211 21,65,82,105,123,133,205,231,265,273,287,451,533,561,689,697,703,861,
%T A020211 1001,1105,1113,1241,1365,1558,1729,1785,1891,2173,2465,2569,2665,2821,
%U A020211 2993,3034,3277,3445,4081,4305,4411,4505,4641,4745,5565,5713,6541,6601
%N A020211 Pseudoprimes to base 83.
%H A020211 Index entries for sequences related to pseudoprimes
%K A020211 nonn
%O A020211 1,1
%A A020211 dww
%I A014641
%S A014641 1,21,65,133,225,341,481,645,833,1045,1281,1541,1825,2133,2465,2821,3201,
%T A014641 3605,4033,4485,4961,5461,5985,6533,7105,7701,8321,8965,9633,10325,11041,
%U A014641 11781,12545,13333,14145,14981,15841,16725,17633,18565,19521,20501,21505
%N A014641 Odd octagonal numbers: (2n+1)(6n+1).
%Y A014641 Cf. A000567, A014642, A014793, A014794.
%K A014641 nonn
%O A014641 0,2
%A A014641 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A014641 More terms from Patrick De Geest (pdg@worldofnumbers.com). Better description from njas.
%I A020309
%S A020309 21,65,231,265,689,703,1241,3445,4411,6973,8421,12871,15883,18721,20191,
%T A020309 22261,24727,31417,42127,44801,60551,69841,79003,82513,83333,85609,92631,
%U A020309 98671,113401,116941,133141,136741,159621,166499,190513,191407,220669
%N A020309 Strong pseudoprimes to base 83.
%H A020309 Index entries for sequences related to pseudoprimes
%K A020309 nonn
%O A020309 1,1
%A A020309 dww
%I A048713
%S A048713 21,65,325,1105,5397,16705,83013,283985,1376277,4259905,
%T A048713 21299525,72418385,353703189,1094795585,5440291909,18611524945,
%U A048713 90194313237,279172874305,1395864371525,4745938863185,23179938501909
%N A048713 3rd row of Family 1 "90 x 150 array": generations 0 .. n of rule 90 starting from seed pattern 21.
%F A048713 SHIFTXORADJ(A048711)
%K A048713 nonn
%O A048713 0,1
%A A048713 Antti.Karttunen@iki.fi (karttu@megabaud.fi).
%I A023712
%S A023712 21,69,81,84,86,87,89,93,101,117,149,213,261,273,276,278,279,
%T A023712 281,285,293,309,321,324,326,327,329,333,336,338,339,344,346,
%U A023712 347,348,350,351,353,356,358,359,361,365,369,372,374,375,377
%N A023712 Exactly 3 1's in base 4 expansion.
%t A023712 Select[ Range[ 400 ], (Count[ IntegerDigits[ #,4 ],1 ]==3)& ]
%K A023712 nonn,base,easy
%O A023712 1,1
%A A023712 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043339
%S A043339 21,69,81,84,86,87,89,93,101,117,149,213,261,273,276,278,279,
%T A043339 281,285,293,309,321,324,326,327,329,333,336,338,339,344,346,
%U A043339 347,348,350,351,353,356,358,359,361,365,369,372,374,375,377
%N A043339 Number of 1's in base 4 is 3.
%K A043339 nonn,base
%O A043339 1,1
%A A043339 Clark Kimberling, ck6@cedar.evansville.edu
%I A045124
%S A045124 21,69,81,84,86,89,101,149,261,273,276,278,281,293,321,324,326,329,336,
%T A045124 338,344,346,353,356,358,361,389,401,404,406,409,421,533,581,593,596,
%U A045124 598,601,613,661,1029,1041,1044,1046,1049,1061
%N A045124 In base 4 representation the numbers of 1's and 3's are 3 and 0, respectively.
%K A045124 nonn,base
%O A045124 1,1
%A A045124 Clark Kimberling, ck6@cedar.evansville.edu
%I A045100
%S A045100 21,69,81,84,87,93,117,213,261,273,276,279,285,309,321,324,327,333,336,
%T A045100 339,348,351,369,372,375,381,453,465,468,471,477,501,789,837,849,852,
%U A045100 855,861,885,981,1029,1041,1044,1047,1053,1077
%N A045100 In base 4 representation the numbers of 1's and 2's are 3 and 0, respectively.
%K A045100 nonn,base
%O A045100 1,1
%A A045100 Clark Kimberling, ck6@cedar.evansville.edu
%I A038471
%S A038471 21,69,81,84,261,273,276,321,324,336,1029,1041,1044,1089,1092,1104,
%T A038471 1281,1284,1296,1344,4101,4113,4116,4161,4164,4176,4353,4356,4368,4416,
%U A038471 5121,5124,5136,5184,5376,16389,16401,16404,16449,16452,16464
%N A038471 Sums of 3 distinct powers of 4.
%Y A038471 Base 4 interpretation of A038445.
%K A038471 nonn,easy
%O A038471 0,1
%A A038471 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A020150
%S A020150 21,69,91,105,161,169,345,483,485,645,805,1105,1183,1247,1261,1541,1649,
%T A020150 1729,1891,2037,2041,2047,2413,2465,2737,2821,3241,3605,3801,5551,5565,
%U A020150 5963,6019,6601,6693,7081,7107,7267,7665,8119,8365,8421,8911,9453,10185
%N A020150 Pseudoprimes to base 22.
%H A020150 Index entries for sequences related to pseudoprimes
%K A020150 nonn
%O A020150 1,1
%A A020150 dww
%I A044159
%S A044159 21,70,119,147,168,217,266,315,364,413,462,490,511,560,609,658,
%T A044159 707,756,805,833,854,903,952,1001,1029,1099,1148,1176,1197,1246,
%U A044159 1295,1344,1393,1442,1491,1519,1540,1589,1638,1687,1736,1785
%N A044159 String 3,0 occurs in the base 7 representation of n but not of n-1.
%K A044159 nonn,base
%O A044159 1,1
%A A044159 Clark Kimberling, ck6@cedar.evansville.edu
%I A044540
%S A044540 21,70,119,153,168,217,266,315,364,413,462,496,511,560,609,658,707,756,
%T A044540 805,839,854,903,952,1001,1077,1099,1148,1182,1197,1246,1295,1344,1393,
%U A044540 1442,1491,1525,1540,1589,1638,1687,1736,1785
%N A044540 String 3,0 occurs in the base 7 representation of n but not of n+1.
%K A044540 nonn,base
%O A044540 1,1
%A A044540 Clark Kimberling, ck6@cedar.evansville.edu
%I A045559
%S A045559 21,78,112,134,135,141,155,182,196,202,206,259,260,266,320,323,330,413,
%T A045559 418,422,428,437,442,463,465,485,490,517,545,548,559,566,572,579,620,
%U A045559 623,650,651,667,680,728,768,781,820,823,833,844
%N A045559 Final 2 nonzero digits of n! are '44'.
%H A045559 Index entries for sequences related to final digits of numbers
%K A045559 nonn
%O A045559 0,1
%A A045559 Jeff Burch (gburch@erols.com)
%I A010009
%S A010009 1,21,78,173,306,477,686,933,1218,1541,1902,2301,2738,3213,
%T A010009 3726,4277,4866,5493,6158,6861,7602,8381,9198,10053,10946,
%U A010009 11877,12846,13853,14898,15981,17102,18261,19458,20693
%N A010009 a(0)=1, a(n)=19*n^2 + 2, n >= 1.
%K A010009 nonn
%O A010009 0,2
%A A010009 njas
%I A064762
%S A064762 0,21,84,189,336,525,756,1029,1344,1701,2100,2541,3024,3549,4116,4725,
%T A064762 5376,6069,6804,7581,8400,9261,10164,11109,12096,13125,14196,15309,
%U A064762 16464,17661,18900,20181,21504,22869,24276,25725,27216,28749
%N A064762 21n^2.
%C A064762 Number of edges in a complete 7-partite graph of order 7n, K_n,n,n,n,n,n,n
%Y A064762 A033583, A033581, A000290, A000217, A033428.
%K A064762 nonn
%O A064762 0,2
%A A064762 Roberto E. Martinez II (remartin@fas.harvard.edu), Oct 18 2001
%I A041856
%S A041856 21,85,106,297,1294,2885,4179,19601,827421,3329285,4156706,
%T A041856 11642697,50727494,113097685,163825179,768398401,32436558021,
%U A041856 130514630485,162951188506,456417007497,1988619218494,4433655444485
%N A041856 Numerators of continued fraction convergents to sqrt(450).
%Y A041856 Cf. A041857.
%K A041856 nonn,cofr,easy
%O A041856 0,1
%A A041856 njas
%I A044208
%S A044208 21,85,149,168,213,277,341,405,469,533,597,661,680,725,789,853,
%T A044208 917,981,1045,1109,1173,1192,1237,1301,1344,1429,1493,1557,1621,
%U A044208 1685,1704,1749,1813,1877,1941,2005,2069,2133,2197,2216,2261
%N A044208 String 2,5 occurs in the base 8 representation of n but not of n-1.
%K A044208 nonn,base
%O A044208 1,1
%A A044208 Clark Kimberling, ck6@cedar.evansville.edu
%I A044589
%S A044589 21,85,149,175,213,277,341,405,469,533,597,661,687,725,789,853,917,981,
%T A044589 1045,1109,1173,1199,1237,1301,1407,1429,1493,1557,1621,1685,1711,1749,
%U A044589 1813,1877,1941,2005,2069,2133,2197,2223,2261
%N A044589 String 2,5 occurs in the base 8 representation of n but not of n+1.
%K A044589 nonn,base
%O A044589 1,1
%A A044589 Clark Kimberling, ck6@cedar.evansville.edu
%I A041858
%S A041858 21,85,361,807,6817,143964,1158529,2461022,11002617,46471490,
%T A041858 1962805197,7897692278,33553574309,75004840896,633592301477,
%U A041858 13380443171913,107677137676781,228734718525475,1022616011778681
%N A041858 Numerators of continued fraction convergents to sqrt(451).
%Y A041858 Cf. A041859.
%K A041858 nonn,cofr,easy
%O A041858 0,1
%A A041858 njas
%I A063651
%S A063651 1,21,85,747,4375,31387,202841,1382259,9167119,61643709,411595537,
%T A063651 2758179839,18448963469,123518353059,826573277157,5532716266089,
%U A063651 37028886137273,247839719105625,1658772577825883,11102227136885119
%N A063651 Ways to tile a 7 X n rectangle with 1 X 1 and 2 X 2 tiles.
%Y A063651 A001045, A054854, A054855, A063650-A063654.
%K A063651 nonn
%O A063651 1,2
%A A063651 Reiner Martin (reinermartin@hotmail.com), Jul 23 2001
%I A045016
%S A045016 21,86,87,89,93,101,117,149,213,346,347,350,351,358,359,361,365,374,
%T A045016 375,377,381,406,407,409,413,421,437,470,471,473,477,485,501,598,599,
%U A045016 601,605,613,629,661,725,854,855,857,861,869,885
%N A045016 In base 4 representation the numbers of 0's and 1's are 0 and 3, respectively.
%K A045016 nonn,base
%O A045016 1,1
%A A045016 Clark Kimberling, ck6@cedar.evansville.edu
%I A020248
%S A020248 21,91,169,485,1183,1247,2047,5551,11557,14111,15229,17767,19909,21667,
%T A020248 23651,38503,47197,49141,53131,64907,70579,72581,90851,100711,109061,
%U A020248 144433,146611,153553,157753,158237,163831,191959,204263,222529,223721
%N A020248 Strong pseudoprimes to base 22.
%H A020248 Index entries for sequences related to pseudoprimes
%K A020248 nonn
%O A020248 1,1
%A A020248 dww
%I A065827
%S A065827 1,21,91,341,651,1911,2451,5461,7381,13671,14763,31031,28731,51471,
%T A065827 59241,87381,83811,155001,130683,221991,223041,310023,280371,496951,
%U A065827 406901,603351,597871,835791,708123,1244061,924483,1398101,1343433
%N A065827 Sum of squares of divisors of square numbers.
%F A065827 Multiplicative with a(p^e) = (p^(4*e+2)-1)/(p^2-1).
%Y A065827 Cf. A001157, A065764.
%K A065827 mult,nonn
%O A065827 1,2
%A A065827 Vladeta Jovovic (vladeta@Eunet.yu), Dec 06 2001
%I A065522
%S A065522 21,93,381,1065,1173,5065,5670,5729,6603,8809,10281,15960,17110,39286,
%T A065522 40526,47882,49951,61962,85058,85261,99066,117860,125985,126853,135890,
%U A065522 143241,171945,179556,185853,208744,209585,210450,251394,261767,288792
%N A065522 Numbers n such that sigma(n) and sigma(n+1) are nontrivial powers (A065496).
%t A065522 a = {}; Do[s = DivisorSigma[1, n]; If[ Position[ Union[ Transpose[ FactorInteger[s]] [[2]]], 1] != {{1}} && Union[ Mod[ Union[ Transpose[ FactorInteger[s]] [[2]]], Union[ Transpose[ FactorInteger[s]] [[2]]] [[1]]]] == {0}, a = Append[a,n]], {n, 2, 10^6} ]; a[[ Select[ Range[ Length[ a]], a[[ # ]] + 1 == a[[ # + 1 ]] & ]]]
%Y A065522 Cf. A065496.
%K A065522 nonn
%O A065522 1,1
%A A065522 Robert G. Wilson v (rgwv@kspaint.com), Nov 27 2001
%I A044272
%S A044272 21,102,183,189,264,345,426,507,588,669,750,831,912,918,993,1074,
%T A044272 1155,1236,1317,1398,1479,1560,1641,1647,1701,1803,1884,1965,
%U A044272 2046,2127,2208,2289,2370,2376,2451,2532,2613,2694,2775,2856
%N A044272 String 2,3 occurs in the base 9 representation of n but not of n-1.
%K A044272 nonn,base
%O A044272 1,1
%A A044272 Clark Kimberling, ck6@cedar.evansville.edu
%I A044653
%S A044653 21,102,183,197,264,345,426,507,588,669,750,831,912,926,993,1074,1155,
%T A044653 1236,1317,1398,1479,1560,1641,1655,1781,1803,1884,1965,2046,2127,2208,
%U A044653 2289,2370,2384,2451,2532,2613,2694,2775,2856
%N A044653 String 2,3 occurs in the base 9 representation of n but not of n+1.
%K A044653 nonn,base
%O A044653 1,1
%A A044653 Clark Kimberling, ck6@cedar.evansville.edu
%I A060537
%S A060537 1,21,105,266,1386,6678,25403,100506,384678,1393903,4831890,15955485,
%T A060537 50080478,149211930,421819950,1132236630,2890927935,7040892159,
%U A060537 16411041500,36733789575,79230165105,165194651065,333926559540
%N A060537 Homeomorphically irreducible multigraphs (or series-reduced multigraphs or multigraphs without nodes of degree 2) on 7 labeled nodes.
%D A060537 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
%H A060537 Vladeta Jovovic, Generating functions for homeomorphically irreducible multigraphs on n labeled nodes
%F A060537 G.f.: (7*x^33 - 42*x^32 + 105*x^31 + 3598*x^30 - 64995*x^29 + 498369*x^28 - 2213029*x^27 + 6169800*x^26 - 10213560*x^25 + 4476990*x^24 + 27664014*x^23 - 97812519*x^22 + 197723150*x^21 - 296237340*x^20 + 352014180*x^19 - 334492361*x^18 + 243984426*x^17 - 117769575*x^16 + 9628325*x^15 + 45726945*x^14 - 50729175*x^13 + 31353175*x^12 - 11717370*x^11 + 1358280*x^10 + 1395765*x^9 - 1068648*x^8 + 395328*x^7 - 77805*x^6 + 882*x^5 + 4095*x^4 - 1141*x^3 + 126*x^2 - 1)/(x - 1)^21. E.g.f. for homeomorphically irreducible multigraphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp(x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k,2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.
%Y A060537 Cf. A003514, A060516, A060533-A060537.
%K A060537 easy,nonn
%O A060537 0,2
%A A060537 Vladeta Jovovic (vladeta@Eunet.yu), Apr 01 2001
%I A041854
%S A041854 21,106,339,445,784,1229,9387,10616,62467,322951,385418,
%T A041854 3020877,3406295,6427172,9833467,35927573,189471332,7993723517,
%U A041854 40158088917,128467990268,168626079185,297094069453,465720148638
%N A041854 Numerators of continued fraction convergents to sqrt(449).
%Y A041854 Cf. A041855.
%K A041854 nonn,cofr,easy
%O A041854 0,1
%A A041854 njas
%I A039611
%S A039611 21,109,153,177,189,201,213,225,237,249,252,254,255,256,257,258,259,
%T A039611 260,262,263,273,285,309,397,453,541,597,685,741,829,885,973,1029,1117,
%U A039611 1173,1261,1297,1308,1310,1311,1312,1313,1314,1315,1316,1318
%N A039611 Representation in base 12 has same nonzero number of 1's and 9's.
%K A039611 nonn,base,easy
%O A039611 0,1
%A A039611 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A039456
%S A039456 21,111,131,153,164,175,186,197,208,219,230,231,233,234,235,236,237,
%T A039456 238,239,240,263,353,384,474,505,595,626,716,747,837,868,958,989,1079,
%U A039456 1110,1200,1211,1221,1223,1224,1225,1226,1227,1228,1229,1230,1233
%N A039456 Representation in base 11 has same nonzero number of 1's and 10's.
%K A039456 nonn,base,easy
%O A039456 0,1
%A A039456 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A054257
%S A054257 21,117,241,283,389,517,547,627,1219,1223,1247,1285,1287,1323,1467,
%T A054257 1505,1591,1599,1689,1713,1817,1961,2203,2309,2377,2607,2837,2847,2899,
%U A054257 2911,3227,3261,3499,3823,4007,4069,4183,4347,4403,4473,4507,4535,4637
%N A054257 Concatenation of n in base 10 down up to base 2 is prime, all numbers are interpreted as decimals.
%e A054257 E.g. a(1)=21 -> 21{10}=23{9}=25{8}=30{7}=33{6}=41{5}=111{4}=210{3}=10101{2} -> 21232530334111121010101 is a prime.
%Y A054257 Cf. A054256, A054258.
%K A054257 nonn,base
%O A054257 0,1
%A A054257 Patrick De Geest (pdg@worldofnumbers.com), Feb 2000.
%I A044353
%S A044353 21,121,210,221,321,421,521,621,721,821,921,1021,1121,1210,1221,
%T A044353 1321,1421,1521,1621,1721,1821,1921,2021,2100,2210,2221,2321,
%U A044353 2421,2521,2621,2721,2821,2921,3021,3121,3210,3221,3321,3421
%N A044353 String 2,1 occurs in the base 10 representation of n but not of n-1.
%K A044353 nonn,base
%O A044353 1,1
%A A044353 Clark Kimberling, ck6@cedar.evansville.edu
%I A044734
%S A044734 21,121,219,221,321,421,521,621,721,821,921,1021,1121,1219,1221,1321,
%T A044734 1421,1521,1621,1721,1821,1921,2021,2199,2219,2221,2321,2421,2521,2621,
%U A044734 2721,2821,2921,3021,3121,3219,3221,3321,3421
%N A044734 String 2,1 occurs in the base 10 representation of n but not of n+1.
%K A044734 nonn,base
%O A044734 1,1
%A A044734 Clark Kimberling, ck6@cedar.evansville.edu
%I A053052
%S A053052 1,21,123,4321,12345,654321,1234567,87654321,123456789,10987654321,
%T A053052 1234567891011,121110987654321,12345678910111213,1413121110987654321,
%U A053052 123456789101112131415
%N A053052 Append n to previous term, reverse alternate terms.
%D A053052 Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000
%H A053052 M. L. Perez et al., eds., Smarandache Notions Journal
%Y A053052 A000422, A007908.
%K A053052 easy,base,nonn
%O A053052 1,2
%A A053052 Felice Russo (felice.russo@katamail.com), Feb 25 2000
%I A002299
%S A002299 1,21,126,462,1287,3003,6188,11628,20349,33649,53130,80730,
%T A002299 118755,169911,237336,324632,435897,575757,749398,962598,
%U A002299 1221759,1533939,1906884,2349060,2869685,3478761,4187106
%N A002299 C(2n+5,5).
%F A002299 G.f. (1+15*x+15*x^2+x^3)/(1-x)^6 = (1+x)*(x^2+14*x+1)/(1-x)^6
%Y A002299 a(n)=A000389(2*n+5).
%Y A002299 Cf. A000389, A053127.
%K A002299 nonn
%O A002299 0,2
%A A002299 njas, Eric Lane (ERICLANE@UTCVM.UTC.EDU)
%I A041852
%S A041852 21,127,5355,32257,1360149,8193151,345472491,2081028097,
%T A041852 87748652565,528572943487,22287812279019,134255446617601,
%U A041852 5661016570218261,34100354867927167,1437875921023159275
%N A041852 Numerators of continued fraction convergents to sqrt(448).
%Y A041852 Cf. A041853.
%K A041852 nonn,cofr,easy
%O A041852 0,1
%A A041852 njas
%I A008384
%S A008384 1,21,131,471,1251,2751,5321,9381,15421,24001,35751,51371,
%T A008384 71631,97371,129501,169001,216921,274381,342571,422751,
%U A008384 516251,624471,748881,891021,1052501,1235001,1440271
%N A008384 Crystal ball sequence for A_4 lattice.
%H A008384 Index entries for crystal ball sequences
%H A008384 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).
%p A008384 35/12*n^4-35/6*n^3+85/12*n^2-25/6*n;
%K A008384 nonn,nice
%O A008384 0,2
%A A008384 njas, J. H. Conway (conway@math.princeton.edu)
%I A033595
%S A033595 1,0,21,136,465,1176,2485,4656,8001,12880,19701,28920,41041,
%T A033595 56616,76245,100576,130305,166176,208981,259560,318801,
%U A033595 387640,467061,558096,661825,779376,911925,1060696,1226961
%N A033595 (n^2-1)*(2*n^2-1).
%K A033595 nonn
%O A033595 0,3
%A A033595 njas
%I A003702 M5110
%S A003702 1,1,21,141,10441,183481,29429661,987318021,276117553681,
%T A003702 15085947275761,6514632269358501,526614587249608701,324871912636292700121,
%U A003702 36433570919762397948841,30417659816002454665514541
%V A003702 1,-1,21,-141,10441,-183481,29429661,-987318021,276117553681,
%W A003702 -15085947275761,6514632269358501,-526614587249608701,324871912636292700121,
%X A003702 -36433570919762397948841,30417659816002454665514541
%N A003702 Expansion of tan(x)/cosh(x).
%t A003702 Tan[ x ]/Cosh[ x ] (* Odd Part *)
%K A003702 sign,done
%O A003702 0,3
%A A003702 R. H. Hardin (rhh@research.bell-labs.com), Simon Plouffe (plouffe@math.uqam.ca)
%I A060493
%S A060493 1,21,147,627,2002,5278,12138,25194,48279,86779,148005,241605,380016,
%T A060493 578956,857956,1240932,1756797,2440113,3331783,4479783,5939934,7776714,
%U A060493 10064110,12886510,16339635,20531511,25583481,31631257,38826012
%N A060493 A diagonal of A060042.
%Y A060493 A060042.
%K A060493 nonn
%O A060493 0,2
%A A060493 Larry Reeves (larryr@acm.org), Mar 20 2001
%I A041850
%S A041850 21,148,6237,43807,1846131,12966724,546448539,3838106497,
%T A041850 161746921413,1136066556388,47876542289709,336271862584351,
%U A041850 14171294770832451,99535335258411508,4194655375624115787
%N A041850 Numerators of continued fraction convergents to sqrt(447).
%Y A041850 Cf. A041851.
%K A041850 nonn,cofr,easy
%O A041850 0,1
%A A041850 njas
%I A022681
%S A022681 1,21,168,511,756,8946,13265,41604,100023,168819,192675,
%T A022681 1687035,551446,9388890,39015,23757153,51335655,33287667,
%U A022681 289673223,168014469,413315910,2158209675,1508351355,6477445065
%V A022681 1,-21,168,-511,-756,8946,-13265,-41604,100023,168819,-192675,
%W A022681 -1687035,551446,9388890,39015,-23757153,-51335655,33287667,
%X A022681 289673223,168014469,-413315910,-2158209675,-1508351355,6477445065
%N A022681 Expansion of Product (1-m*q^m)^21; m=1..inf.
%K A022681 sign,done
%O A022681 0,2
%A A022681 njas
%I A041848
%S A041848 21,169,359,887,1246,4625,5871,10496,215791,226287,442078,
%T A041848 1552521,1994599,5541719,13078037,110166015,4640050667,
%U A041848 37230571351,79101193369,195432958089,274534151458,1019035412463
%N A041848 Numerators of continued fraction convergents to sqrt(446).
%Y A041848 Cf. A041849.
%K A041848 nonn,cofr,easy
%O A041848 0,1
%A A041848 njas
%I A007261 M5111
%S A007261 1,21,171,745,2418,7587,20510,51351,122715,277384,598812,1255761,
%T A007261 2543973
%N A007261 McKay-Thompson series of class 6b for Monster.
%D A007261 J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
%D A007261 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%D A007261 J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18 (1990), no. 1, 253-278.
%K A007261 nonn
%O A007261 0,2
%A A007261 njas
%I A015880
%S A015880 21,174,270,517,572,913,992,1002,1420,1633,1830,2622,2958,4170,4747,
%T A015880 5539,7520,7544,7729,10184,10783,14863,16165,16520,19837,20935,
%U A015880 21584,23161,26840,28544,29737,31453,34510,35571,35611,35845,39560
%N A015880 sigma(n) = sigma(n + 10).
%K A015880 nonn
%O A015880 0,1
%A A015880 Robert G. Wilson v (rgwv@kspaint.com)
%I A025604
%S A025604 1,0,21,180,3699,78186,1763689,40219584,921030783,21114454518,
%T A025604 484210273013,11105346505100,254708804007115,5841980066693506,
%U A025604 133991588716915713
%N A025604 n-move queen paths on 8x8 board from given corner to same corner.
%K A025604 nonn
%O A025604 0,3
%A A025604 dww
%I A062159
%S A062159 1,0,21,182,819,2604,6665,14706,29127,53144,90909,147630,229691,344772,
%T A062159 501969,711914,986895,1340976,1790117,2352294,3047619,3898460,4929561,
%U A062159 6168162,7644119,9390024,11441325,13836446,16616907,19827444,23516129
%V A062159 -1,0,21,182,819,2604,6665,14706,29127,53144,90909,147630,229691,344772,501969,711914,
%W A062159 986895,1340976,1790117,2352294,3047619,3898460,4929561,6168162,7644119,9390024,
%X A062159 11441325,13836446,16616907,19827444,23516129,27734490,32537631,37984352,44137269
%N A062159 n^5-n^4+n^3-n^2+n-1.
%F A062159 a(n) =round[n^6/(n+1)] for n>2 =A062160(n,6).
%e A062159 a(4) =4^5-4^4+4^3-4^2+4-1 =1024-256+64-16+4-1 =819
%Y A062159 Cf. A023443, A002061, A062158, A060884, A060888.
%K A062159 easy,sign
%O A062159 0,3
%A A062159 Henry Bottomley (se16@btinternet.com), Jun 08 2001
%I A059721
%S A059721 0,1,21,182,910,3255,9331,22876,49932,99645,185185,324786,542906,
%T A059721 871507,1351455,2034040,2982616,4274361,6002157,8276590,11228070,
%U A059721 15009071,19796491,25794132,33235300,42385525,53545401,67053546
%N A059721 Mean of first six positive powers of n, i.e. (n+n^2+n^3+n^4+n^5+n^6)/6.
%F A059721 a(n) =(n^7-n)/(6n-6) =A053700(n)*n/6
%e A059721 a(2)=(2+4+8+16+32+64)/6=126/6=21
%Y A059721 Cf. A059722, A059723.
%K A059721 nonn
%O A059721 0,3
%A A059721 Henry Bottomley (se16@btinternet.com), Feb 07 2001
%I A054370
%S A054370 1,1,0,21,182,2093,23394,285383,3621150,47813367,650302814
%N A054370 Unlabeled asymmetric 7-ary cacti having n polygons.
%H A054370 Miklos Bona, Michel Bousquet, Gilbert Labelle, Pierre Leroux, Enumeration of m-ary cacti, Advances in Applied Mathematics, 24 (2000), 22-56 (pdf, dvi).
%H A054370 Index entries for sequences related to cacti
%K A054370 nonn
%O A054370 0,4
%A A054370 Simon Plouffe (plouffe@math.uqam.ca)
%I A010827
%S A010827 1,21,189,910,2205,378,13321,33345,10395,86870,122703,46683,
%T A010827 98287,264915,96390,1163064,1113588,1066527,1042055,536025,
%U A010827 2287467,3603805,1391733,478170,562555,13742379,7889805
%V A010827 1,-21,189,-910,2205,-378,-13321,33345,-10395,-86870,122703,46683,
%W A010827 -98287,-264915,96390,1163064,-1113588,-1066527,1042055,536025,
%X A010827 2287467,-3603805,-1391733,478170,-562555,13742379,-7889805
%N A010827 Expansion of Product (1-x^k )^21.
%D A010827 Newman, Morris; A table of the coefficients of the powers of $\eta(\tau)$. Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
%K A010827 sign,done
%O A010827 0,2
%A A010827 njas
%I A022713
%S A022713 1,21,189,952,3087,8400,27503,92907,246981,588490,1649823,
%T A022713 4645179,10989769,25476717,65615424,161705810,363998754,
%U A022713 834169539,1973130271,4510112565,9969930495,22054706773
%V A022713 1,-21,189,-952,3087,-8400,27503,-92907,246981,-588490,1649823,
%W A022713 -4645179,10989769,-25476717,65615424,-161705810,363998754,
%X A022713 -834169539,1973130271,-4510112565,9969930495,-22054706773
%N A022713 Expansion of Product (1+m*q^m)^-21; m=1..inf.
%K A022713 sign,done
%O A022713 0,2
%A A022713 njas
%I A027780
%S A027780 21,196,1008,3780,11550,30492,72072,156156,315315,600600,1089088,
%T A027780 1893528,3174444,5155080,8139600,12534984,18877089,27861372,
%U A027780 40378800,57557500,80810730,111891780,152956440,206633700
%N A027780 7*(n+1)*C(n+2,7)/2.
%F A027780 Number of 10-subsequences of [ 1,n ] with just 2 contiguous pairs; g.f. 7*(3+x)/(1-x)^9
%K A027780 nonn
%O A027780 0,1
%A A027780 thi ngoc dinh (via rkg@cpsc.ucalgary.ca)
%I A058086
%S A058086 1,21,196,2100,23310,277662,3541188,48323988,703842951,10911044795,
%T A058086 179505659256,3125293564568,57431191568268,1111135121304012,22580910916230360,
%U A058086 480992394437882040,10717471923433878141,249346024282594457841
%N A058086 Coefficients of menage hit polynomials.
%D A058086 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.
%Y A058086 A diagonal of A058057.
%K A058086 nonn
%O A058086 6,2
%A A058086 njas, Dec 02 2000
%I A036205
%S A036205 0,21,208,42,98,1,107,63,188,119,162,22,117,128,78,84,60,
%T A036205 209,92,140,87,183,109,43,196,138,168,149,145,99,29,105,
%U A036205 142,81,205,2,220,113,97,161,59,108,144,204,58,130,131
%N A036205 Log base 6 (n) mod 229.
%D A036205 I. M. Vinogradov, Elements of Number Theory, p. 220ff
%K A036205 nonn,fini
%O A036205 1,2
%A A036205 njas
%I A047646
%S A047646 1,21,210,1330,5964,19929,50253,91920,97965,51604,526659,1389297,
%T A047646 2280320,2118690,769065,7613319,17220042,23999430,18024405,10748850,
%U A047646 63778953,124134772,152793270,99072120,71722224,341062407,610085721
%V A047646 1,-21,210,-1330,5964,-19929,50253,-91920,97965,51604,-526659,1389297,
%W A047646 -2280320,2118690,769065,-7613319,17220042,-23999430,18024405,10748850,
%X A047646 -63778953,124134772,-152793270,99072120,71722224,-341062407,610085721
%N A047646 Expand {Product_{j=1..inf} (1-x^j) - 1 }^21 in powers of x.
%D A047646 H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
%K A047646 sign,done
%O A047646 1,2
%A A047646 njas
%I A010937
%S A010937 1,21,210,1330,5985,20349,54264,116280,203490,293930,352716,
%T A010937 352716,293930,203490,116280,54264,20349,5985,1330,210,
%U A010937 21,1
%N A010937 Binomial coefficient C(21,n).
%K A010937 nonn,fini,full
%O A010937 0,2
%A A010937 njas
%I A022616
%S A022616 1,21,210,1351,6426,24780,82845,250806,703731,1853481,4628337,
%T A022616 11052867,25403952,56451192,121738767,255623851,524037507,
%U A022616 1051143723,2066899387,3990768663,7577013360,14163858895
%V A022616 1,-21,210,-1351,6426,-24780,82845,-250806,703731,-1853481,4628337,
%W A022616 -11052867,25403952,-56451192,121738767,-255623851,524037507,
%X A022616 -1051143723,2066899387,-3990768663,7577013360,-14163858895
%N A022616 Expansion of Product (1+q^m)^-21; m=1..inf.
%K A022616 sign,done
%O A022616 0,2
%A A022616 njas
%I A041846
%S A041846 21,211,232,443,4662,196247,1967132,2163379,4130511,43468489,
%T A041846 1829807049,18341538979,20171346028,38512885007,405300196098,
%U A041846 17061121121123,171016511407328,188077632528451,359094143935779
%N A041846 Numerators of continued fraction convergents to sqrt(445).
%Y A041846 Cf. A041847.
%K A041846 nonn,cofr,easy
%O A041846 0,1
%A A041846 njas
%I A053063
%S A053063 1,21,213,4213,42135,642135,6421357,86421357,864213579,10864213579,
%T A053063 1086421357911,121086421357911,12108642135791113
%N A053063 Alternately append n to beginning or end of previous term.
%D A053063 Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000
%H A053063 M. L. Perez et al., eds., Smarandache Notions Journal
%K A053063 easy,nonn
%O A053063 1,2
%A A053063 Felice Russo (felice.russo@katamail.com), Feb 25 2000
%I A060933
%S A060933 1,21,217,1498,7910,34566,131446,449732,1416513,4174765,11651717,
%T A060933 31075422,79751854,198036146,477899790,1124785648,2589534248,
%U A060933 5845989156,12968091584,28316428700,60953528230
%N A060933 Sixth convolution of Lucas numbers A000032(n+1), n >= 0.
%F A060933 a(n)= A060922(n+6,6) (seventh column of Lucas triangle).
%F A060933 G.f.: ((1+2*x)/(1-x-x^2))^6.
%F A060933 a(n)= (n+1)*(2*(100*n^5+845*n^4+2480*n^3+4345*n^2+5910*n+2952)*L(n+2) + (125*n^5+1030*n^4+2995*n^3+5930*n^2+8280*n+288)*L(n+1))/(6!*5^2), with the Lucas numbers L(n)=A000032(n)=A000204(n), n >= 1.
%Y A060933 A004799(n+1)= A060922(n+1, 1), A060929-A060932.
%K A060933 nonn,easy
%O A060933 0,2
%A A060933 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Apr 20 2001
%I A008421
%S A008421 1,21,221,1561,8361,36365,134245,433905,1256465,3317445,
%T A008421 8097453,18474633,39753273,81270333,158819253,298199265,
%U A008421 540279585,948062325,1616336765,2684641785,4354393801
%N A008421 Crystal ball sequence for 10-dimensional cubic lattice.
%H A008421 Index entries for crystal ball sequences
%H A008421 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).
%F A008421 G.f.: (1+x)^10/(1-x)^11.
%K A008421 nonn,easy
%O A008421 0,2
%A A008421 njas
%I A027812
%S A027812 21,224,1260,5040,16170,44352,108108,240240,495495,960960,1769768,
%T A027812 3118752,5290740,8682240,13837320,21488544,32605881,48454560,
%U A027812 70662900,101301200,142972830,198918720,273136500,370515600,496989675
%N A027812 7*(n+1)*C(n+5,7).
%F A027812 Number of 13-subsequences of [ 1,n ] with just 5 contiguous pairs; g.f. 7*(3+5x)/(1-x)^9
%K A027812 nonn
%O A027812 0,1
%A A027812 thi ngoc dinh (via rkg@cpsc.ucalgary.ca)
%I A027508
%S A027508 1,21,227,4283,3303383,4156965497,2439821215787,25768363425203859,
%T A027508 2832420937806122764581,4560420339293289548844701,
%U A027508 1405739820979794860218217874663237
%N A027508 Second column of A027495.
%F A027508 Numerators of sequence a[ 2,n ] in (a[ i,j ])^2 where a[ i,j ] = S(i,j)/BellNumber(i) if j<=i, 0 if j>i
%K A027508 nonn
%O A027508 2,2
%A A027508 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A020267
%S A020267 21,231,671,703,841,1281,1387,1417,2701,3829,8321,8911,10933,13019,14091,
%T A020267 20591,21667,25909,43213,43273,45221,45449,47197,48133,51319,60551,74089,
%U A020267 76861,79381,80137,84941,86331,91669,106491,109981,110293,111361,113401
%N A020267 Strong pseudoprimes to base 41.
%H A020267 Index entries for sequences related to pseudoprimes
%K A020267 nonn
%O A020267 1,1
%A A020267 dww
%I A064322
%S A064322 0,1,21,231,1540,7260,26796,82621,222111,536130,1186570,2445366,
%T A064322 4747821,8763391,15487395,26357430,43398586,69401871,108140571,
%U A064322 164629585,245433090,359026206,516216646,730632651,1019283825
%N A064322 Triply triangular numbers.
%F A064322 a(n) = A000217(A000217(A000217(n))) = n*(n+1)*(n^2+n+2)*(n^4+2n^3+n^2+2n^2+2n+8)/128 = A002817(n)*(A002817(n)+1)/2
%e A064322 a(4)=1540 because 4th triangular number is 10, 10th triangular number is 55, and 55th triangular number is 1540.
%K A064322 nonn
%O A064322 0,3
%A A064322 Henry Bottomley (se16@btinternet.com), Oct 15 2001
%I A010973
%S A010973 1,21,231,1771,10626,53130,230230,888030,3108105,10015005,
%T A010973 30045015,84672315,225792840,573166440,1391975640,3247943160,
%U A010973 7307872110,15905368710,33578000610,68923264410,137846528820
%N A010973 Binomial coefficient C(n,20).
%K A010973 nonn
%O A010973 20,2
%A A010973 njas
%I A022586
%S A022586 1,21,231,1792,11067,58002,268093,1120899,4315269,15497986,
%T A022586 52441347,168487473,517184185,1524390777,4332440454,11914441196,
%U A022586 31798680774,82574231187,209091601271,517272712845,1252351944165
%N A022586 Expansion of Product (1+q^m)^21; m=1..inf.
%K A022586 nonn
%O A022586 0,2
%A A022586 njas
%I A023019
%S A023019 1,21,252,2233,16170,100926,560945,2837418,13266099,57994475,239170239,
%T A023019 937026279,3507380170,12601619226,43628951025,146036139347,473924014599,
%U A023019 1494785958435,4591920193357,13764656869425,40328218603134
%N A023019 Partitions of n into parts of 21 kinds.
%K A023019 nonn
%O A023019 0,2
%A A023019 dww
%I A036220
%S A036220 1,21,252,2268,17010,112266,673596,3752892,19702683,98513415,472864392,
%T A036220 2192371272,9865670724,43257171636,185387878440,778629089448,
%U A036220 3211844993973,13036312034361,52145248137444,205836505805700
%N A036220 Expansion of 1/(1-3*x)^7; 7-fold convolution of A000244 (powers of 3).
%C A036220 a(n)=A027465(n+7,7) (O. Gerard's triangle).
%F A036220 a(n) = 3^n*binomial(n+6,6); G.f. 1/(1-3*x)^7.
%Y A036220 A000244, A027465.
%K A036220 easy,nonn
%O A036220 0,2
%A A036220 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A022649
%S A022649 1,21,252,2275,17052,111594,657419,3557256,17931459,85093393,
%T A022649 383211885,1648114587,6803773998,27073141326,104199172863,
%U A022649 389047640905,1412682672765,4999550830281,17277525329491
%N A022649 Expansion of Product (1+m*q^m)^21; m=1..inf.
%K A022649 nonn
%O A022649 0,2
%A A022649 njas
%I A045505
%S A045505 1,21,262,2525,20754,152946,1040556,6659037,40599130,237978598,
%T A045505 1350216660,7453221490,40188242420,212349718980,1102352779992,
%U A045505 5634083759325,28400234400810,141402315307550,696257439473860
%N A045505 Convolution of A000108 (Catalan numbers) with A040075.
%C A045505 Also convolution of A045492 with A000984 (central binomial coefficients); also convolution of A042985 with A000302 (powers of 4).
%F A045505 a(n) = binomial(n+5,4)*(4^(n+1) - A000984(n+5)/A000984(4))/2, A000984(n)=binomial(2*n,n); G.f. c(x)/(1-4*x)^5, c(x): G.f. Catalan numbers.
%K A045505 easy,nonn
%O A045505 0,2
%A A045505 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A056282
%S A056282 0,0,0,0,0,1,21,266,2646,22827,179487,1323651,9321312,63436352,
%T A056282 420693273,2734926292,17505749898,110687248392,693081601779,
%U A056282 4306078872557,26585679462783,163305339165738,998969857983405
%N A056282 Primitive (aperiodic) word structures of length n which contain exactly six different characters.
%C A056282 Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.
%D A056282 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%F A056282 sum mobius(d)*A000770(n/d) where d|n and n>0.
%Y A056282 Cf. A056271.
%K A056282 nonn
%O A056282 1,7
%A A056282 Marks R. Nester (nesterm@qfri1.se2.dpi.qld.gov.au)
%I A000770 M5112 N2215
%S A000770 1,21,266,2646,22827,179487,1323652,9321312,63436373,420693273,
%T A000770 2734926558,17505749898,110687251039,693081601779,4306078895384
%N A000770 Stirling numbers of second kind.
%D A000770 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
%D A000770 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
%H A000770 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 349
%F A000770 G.f.: x^6/product(1-k*x,k=1..6). E.g.f. ((exp(x)-1)^6)/6!.
%Y A000770 a(n)= A008277(n, 6) (Stirling2 triangle).
%K A000770 nonn
%O A000770 6,2
%A A000770 njas
%I A032535
%S A032535 1,21,273,2231,10101,28261,611123,1200341,3427673,8108919,38636301,
%T A032535 51484647
%N A032535 Odd decimal 'base 2 looking' numbers divided by their actual base 2 values (n) is an integer.
%Y A032535 = odd(A032533). See also A032532 for explanation.
%K A032535 nonn
%O A032535 0,2
%A A032535 Patrick De Geest (pdg@worldofnumbers.com), april 1998.
%I A022745
%S A022745 1,21,273,2716,22659,165984,1098615,6695559,38085117,204218630,
%T A022745 1040291595,5064987207,23686610269,106828575357,466231753944,
%U A022745 1974651627802,8136148603086,32681975601387,128221943065839
%N A022745 Expansion of Product (1-m*q^m)^-21; m=1..inf.
%K A022745 nonn
%O A022745 0,2
%A A022745 njas
%I A004324
%S A004324 1,21,276,2925,27405,237336,1947792,15380937,118030185,
%T A004324 886163135,6540715896,47626016970,343006888770,2448296039700,
%U A004324 17345898649800,122131734269895,855420636763836,5964720367660956
%N A004324 Binomial coefficient C(3n,n-6).
%D A004324 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.
%K A004324 nonn,easy
%O A004324 6,2
%A A004324 njas
%I A028053
%S A028053 1,21,280,3030,29071,258111,2172430,17592960,138433141,
%T A028053 1065579801,8062670980,60182175690,444345657211,3251891531091,
%U A028053 23627323915930,170652039023220,1226510307989281,8779155519807981
%N A028053 Expansion of 1/((1-3x)(1-5x)(1-6x)(1-7x)).
%K A028053 nonn
%O A028053 0,2
%A A028053 njas
%I A028033
%S A028033 1,21,283,3129,31003,287217,2547091,21931833,185027755,
%T A028033 1538650113,12663786499,103454051337,840560481307,6802315926609,
%U A028033 54886579262707,441905197121241,3552134005816459,28518494951667105
%N A028033 Expansion of 1/((1-3x)(1-4x)(1-6x)(1-8x)).
%K A028033 nonn
%O A028033 0,2
%A A028033 njas
%I A025987
%S A025987 1,21,285,3185,31941,299481,2685565,23352945,198684981,
%T A025987 1663903241,13774041645,113050606305,921961387621,7483064823801,
%U A025987 60518933442525,488128819261265,3929148977523861,31579173926461161
%N A025987 Expansion of 1/((1-2x)(1-5x)(1-6x)(1-8x)).
%K A025987 nonn
%O A025987 0,2
%A A025987 njas
%I A028028
%S A028028 1,21,286,3234,33187,322455,3035152,28040628,256229413,
%T A028028 2326373049,21042916258,189930286182,1712158289479,15423616827003,
%U A028028 138884571944404,1250325501927096,11254768340210185,101302177508279517
%N A028028 Expansion of 1/((1-3x)(1-4x)(1-5x)(1-9x)).
%K A028028 nonn
%O A028028 0,2
%A A028028 njas
%I A025971
%S A025971 1,21,287,3249,33159,317457,2913079,25951233,226266887,
%T A025971 1941258033,16450020951,138048250977,1149556141735,9512774280849,
%U A025971 78317007130103,642041894674881,5244884610706503,42718957418411505
%N A025971 Expansion of 1/((1-2x)(1-4x)(1-7x)(1-8x)).
%K A025971 nonn
%O A025971 0,2
%A A025971 njas
%I A025967
%S A025967 1,21,289,3321,34705,343257,3282913,30740457,283960369,
%T A025967 2599944633,23667405697,214622461833,1941330570193,17530479823449,
%U A025967 158125883415841,1425244488222249,12839878208747377,115635005126816505
%N A025967 Expansion of 1/((1-2x)(1-4x)(1-6x)(1-9x)).
%K A025967 nonn
%O A025967 0,2
%A A025967 njas
%I A022452
%S A022452 1,21,290,3330,34491,334791,3109900,27997860,246301781,
%T A022452 2129087961,18155626710,153166509990,1281087729871,10640517267531,
%U A022452 87874475534720,722286313011720,5913514800094761,48255642147081501
%N A022452 Expansion of 1/((1-x)(1-5x)(1-7x)(1-8x)).
%K A022452 nonn
%O A022452 0,2
%A A022452 njas
%I A022291
%S A022291 1,21,292,3402,36043,360843,3485854,32899944,305751325,
%T A022291 2812114305,25683350056,233457113526,2115260975647,19123756383207,
%U A022291 172639882457698,1556953539144948,14031940169321809,126404565100316349
%N A022291 Expansion of 1/((1-x)(1-5x)(1-6x)(1-9x)).
%K A022291 nonn
%O A022291 0,2
%A A022291 njas
%I A025944
%S A025944 1,21,292,3414,36439,368607,3604114,34449888,324158197,
%T A025944 3016246233,27838148416,255387389322,2332396465075,21228942680499,
%U A025944 192722119904398,1746130572580116,15796595736313873,142739306095633005
%N A025944 Expansion of 1/((1-2x)(1-3x)(1-7x)(1-9x)).
%K A025944 nonn
%O A025944 0,2
%A A025944 njas
%I A025962
%S A025962 1,21,293,3465,37821,396081,4058293,41102985,413760941,
%T A025962 4151788641,41590878693,416282134905,4164721639261,41656852053201,
%U A025962 416617235689493,4166418080133225,41665418007392781,416660400463837761
%N A025962 Expansion of 1/((1-2x)(1-4x)(1-5x)(1-10x)).
%K A025962 nonn
%O A025962 0,2
%A A025962 njas
%I A027474
%S A027474 1,21,294,3430,36015,352947,3294172,29647548,259416045,2219448385,
%T A027474 18643366434,154231485954,1259557135291
%N A027474 Third column of A027466.
%F A027474 Numerators of sequence a[ 3,n ] in (a[ i,j ])^3 where a[ i,j ] = Binomial(i-1,j-1)/2^(i-1) if j<=i, 0 if j>i
%K A027474 nonn
%O A027474 3,2
%A A027474 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A021864
%S A021864 1,21,294,3466,37275,379407,3727648,35761572,337430709,
%T A021864 3146261833,29083452762,267126341118,2441786303503,22239710844099,
%U A021864 202003194298836,1830950757258904,16569016601938857,149754390830203005
%N A021864 Expansion of 1/((1-x)(1-4x)(1-7x)(1-9x)).
%K A021864 nonn
%O A021864 0,2
%A A021864 njas
%I A020570
%S A020570 1,21,295,3465,36751,365001,3463615,31794105,284628751,
%T A020570 2499039081,21606842335,184519243545,1559982264751,13079717026761,
%U A020570 108915112739455,901732722577785,7429565635164751,60963378722560041
%N A020570 Expansion of 1/((1-6x)(1-7x)(1-8x)).
%K A020570 nonn
%O A020570 0,2
%A A020570 njas
%I A025940
%S A025940 1,21,295,3525,38911,411741,4255255,43385925,438878671,
%T A025940 4418961261,44370835015,444796211925,4454490876031,44584086067581,
%U A025940 446075938835575,4462169900295525,44630162203602991,446352401627074701
%N A025940 Expansion of 1/((1-2x)(1-3x)(1-6x)(1-10x)).
%K A025940 nonn
%O A025940 0,2
%A A025940 njas
%I A041844
%S A041844 21,295,12411,174049,7322469,102688615,4320244299,60586108801,
%T A041844 2548936813941,35745701503975,1503868399980891,21089903301236449,
%U A041844 887279807051911749,12443007202028000935,523493582292227951019
%N A041844 Numerators of continued fraction convergents to sqrt(444).
%Y A041844 Cf. A041845.
%K A041844 nonn,cofr,easy
%O A041844 0,1
%A A041844 njas
%I A021829
%S A021829 1,21,297,3577,39753,422793,4384969,44813769,454009545,
%T A021829 4575676105,45971643081,461011315401,4617904831177,46225887853257,
%U A021829 462540273695433,4627092539587273,46281069938293449,462871589543951049
%N A021829 Expansion of 1/((1-x)(1-4x)(1-6x)(1-10x)).
%K A021829 nonn
%O A021829 0,2
%A A021829 njas
%I A019943
%S A019943 1,21,298,3570,38971,401751,3988468,38583300,366449941,
%T A019943 3434404281,31873887838,293663563830,2690806228111,24553315831611,
%U A019943 223338364450408,2026585451393160,18355202849805481,166009125098571741
%N A019943 Expansion of 1/((1-5x)(1-7x)(1-9x)).
%K A019943 nonn
%O A019943 0,2
%A A019943 njas
%I A021634
%S A021634 1,21,298,3594,39763,417567,4236796,41963988,408348325,
%T A021634 3920543913,37248255694,350941450782,3284131050487,30562445793459,
%U A021634 283104150028192,2612274474873576,24025167335391049,220344082704460605
%N A021634 Expansion of 1/((1-x)(1-3x)(1-8x)(1-9x)).
%K A021634 nonn
%O A021634 0,2
%A A021634 njas
%I A021604
%S A021604 1,21,300,3670,41511,449151,4730890,48987840,501640821,
%T A021604 5098774681,51564400680,519680849610,5225067192931,52448485198611,
%U A021604 525869552045670,5268388442396980,52751734942121841,527992303241992941
%N A021604 Expansion of 1/((1-x)(1-3x)(1-7x)(1-10x)).
%K A021604 nonn
%O A021604 0,2
%A A021604 njas
%I A025935
%S A025935 1,21,300,3710,43071,485751,5405170,59772720,659098341,
%T A025935 7258131881,79879876440,878881296930,9668709132811,106360879560411,
%U A025935 1169995084978110,12870073026808340,141571438884146481
%N A025935 Expansion of 1/((1-2x)(1-3x)(1-5x)(1-11x)).
%K A025935 nonn
%O A025935 0,2
%A A025935 njas
%I A007592 M5113
%S A007592 21,301,325,697,1333,1909,2041,2133,3901,10693,16513,19521,24601,
%T A007592 26977,51301,96361,130153,159841,163201,176661,214273,250321,275833,
%U A007592 296341,306181,389593,486877,495529,542413,808861,1005421,1005649,1055833,1063141,1232053
%N A007592 Hyperperfect numbers: n=m(sigma(n)-n-1)+1 for some m>1.
%D A007592 D. Minoli, Issues in non-linear hyperperfect numbers, Math. Comp., 34 (1980), 639-645.
%D A007592 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.
%H A007592 J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3
%H A007592 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A007592 See A034897 (for m >= 1).
%K A007592 nonn,nice
%O A007592 1,1
%A A007592 njas
%E A007592 More terms from jud.mccranie@mindspring.com (Jud Mccranie) 10/97.
%I A019664
%S A019664 1,21,301,3669,40957,433125,4418317,43942773,428973853,
%T A019664 4128937989,39306876973,370937567637,3475860284989,32382187083813,
%U A019664 300235508341069,2772487245505461,25515330868003165,234141560259529797
%N A019664 Expansion of 1/((1-4x)(1-8x)(1-9x)).
%K A019664 nonn
%O A019664 0,2
%A A019664 njas
%I A019839
%S A019839 1,21,301,3681,41461,445641,4658221,47871201,486836581,
%T A019839 4919066361,49504632541,496978967121,4981629662101,49888557269481,
%U A019839 499325240101261,4995920923029441,49975372950286021,499851474762263001
%N A019839 Expansion of 1/((1-5x)(1-6x)(1-10x)).
%K A019839 nonn
%O A019839 0,2
%A A019839 njas
%I A021784
%S A021784 1,21,302,3762,43923,497223,5545264,61398804,677478725,
%T A021784 7463074905,82149266706,903924739926,9944608539607,109397965416267,
%U A021784 1203414334895828,13237742692094328,145616100380861769
%N A021784 Expansion of 1/((1-x)(1-4x)(1-5x)(1-11x)).
%K A021784 nonn
%O A021784 0,2
%A A021784 njas
%I A019618
%S A019618 1,21,303,3745,42711,464961,4918663,51086385,524227671,
%T A019618 5336085601,54018566823,544793838225,5480212349431,55028108373441,
%U A019618 551863246323783,5529708675105265,55374624529091991,554289026917064481
%N A019618 Expansion of 1/((1-4x)(1-7x)(1-10x)).
%K A019618 nonn
%O A019618 0,2
%A A019618 njas
%I A021524
%S A021524 1,21,304,3822,45031,513639,5760910,64038576,708445573,
%T A021524 7817058249,86132670988,948329828082,10436851589347,114836710756971,
%U A021524 1263391885146058,13898439159046260,152889601348716673
%N A021524 Expansion of 1/((1-x)(1-3x)(1-6x)(1-11x)).
%K A021524 nonn
%O A021524 0,2
%A A021524 njas
%I A021268
%S A021268 1,21,305,3825,44481,494721,5346625,56661825,592183361,
%T A021268 6126355521,62899732545,642086748225,6525582872641,66093551865921,
%U A021268 667637303808065,6729987319337025,67728787443552321,680719188437241921
%N A021268 Expansion of 1/((1-x)(1-2x)(1-8x)(1-10x)).
%K A021268 nonn
%O A021268 0,2
%A A021268 njas
%I A018069
%S A018069 1,21,307,3873,45235,504633,5465323,58007361,606913219,
%T A018069 6283868745,64556638939,659310178449,6703052628403,67910134629657,
%U A018069 686138217844555,6917677165178337,69627131588692387,699874195511336169
%N A018069 Expansion of 1/((1-3x)(1-8x)(1-10x)).
%K A018069 nonn
%O A018069 0,2
%A A018069 njas
%I A019488
%S A019488 1,21,307,3897,46243,529953,5961259,66380889,735097555,
%T A019488 8115781905,89452902331,985061928201,10842178002787,119303005894977,
%U A019488 1312567620466123,14439652232597433,158844629298359539
%N A019488 Expansion of 1/((1-4x)(1-6x)(1-11x)).
%K A019488 nonn
%O A019488 0,2
%A A019488 njas
%I A025929
%S A025929 1,21,307,3969,48979,593817,7152139,85937313,1031713507,
%T A025929 12382483113,148597656571,1783203843057,21398575559635,
%U A025929 256783429254009,3081403255517803,36976847527073601,443722204297332163
%N A025929 Expansion of 1/((1-2x)(1-3x)(1-4x)(1-12x)).
%K A025929 nonn
%O A025929 0,2
%A A025929 njas
%I A021244
%S A021244 1,21,308,3942,47271,547407,6213586,69694464,776054741,
%T A021244 8602512633,95089014384,1049208790266,11563904125411,127361197423299,
%U A021244 1402080935995502,15430644646390548,169791371563507281
%N A021244 Expansion of 1/((1-x)(1-2x)(1-7x)(1-11x)).
%K A021244 nonn
%O A021244 0,2
%A A021244 njas
%I A018054
%S A018054 1,21,310,3990,48031,557571,6338620,71164380,792891661,
%T A018054 8792412321,97210822930,1072779241170,11824793506891,130242283148271,
%U A018054 1433852001421240,15780680237514360,173645640208869721
%N A018054 Expansion of 1/((1-3x)(1-7x)(1-11x)).
%K A018054 nonn
%O A018054 0,2
%A A018054 njas
%I A021484
%S A021484 1,21,310,4050,50371,613671,7411240,89174100,1071295141,
%T A021484 12861600921,154369595770,1852587338550,22231809806311,
%U A021484 266785528785771,3201445408153900,38417440232993400,461009759536223881
%N A021484 Expansion of 1/((1-x)(1-3x)(1-5x)(1-12x)).
%K A021484 nonn
%O A021484 0,2
%A A021484 njas
%I A016321
%S A016321 1,21,313,4065,49081,566721,6350473,69654225,751887961,
%T A016321 8016991521,84652923433,886876310385,9231886792441,95586981129921,
%U A016321 985282830165193,10117545471478545,103557909243290521,1057021183189581921
%N A016321 Expansion of 1/((1-2x)(1-9x)(1-10x)).
%K A016321 nonn
%O A016321 0,2
%A A016321 njas
%I A019041
%S A019041 1,21,313,4125,51601,630741,7630633,91892685,1104403201,
%T A019041 13261555461,159183299353,1910426955645,22926277062001,
%U A019041 275121159824181,3301483361726473,39617948633641005,475416129363276001
%N A019041 Expansion of 1/((1-4x)(1-5x)(1-12x)).
%K A019041 nonn
%O A019041 0,2
%A A019041 njas
%I A021214
%S A021214 1,21,313,4137,51961,637497,7733881,93310329,1122747001,
%T A021214 13491103353,162002078329,1944677972601,23340053875321,
%U A021214 280104155744889,3361390924417657,40337537425951353,484055527109184121
%N A021214 Expansion of 1/((1-x)(1-2x)(1-6x)(1-12x)).
%K A021214 nonn
%O A021214 0,2
%A A021214 njas
%I A016318
%S A016318 1,21,315,4145,51051,605241,7007155,79874865,900993051,
%T A016318 10089880361,112420339395,1248076978785,13820472734251,
%U A016318 152758207825881,1686204348094035,18595160325141905,204922063545486651
%N A016318 Expansion of 1/((1-2x)(1-8x)(1-11x)).
%K A016318 nonn
%O A016318 0,2
%A A016318 njas
%I A017954
%S A017954 1,21,315,4185,52731,648081,7869555,94992345,1143260811,
%T A017954 13739265441,164992058595,1980630120105,23771914474491,
%U A017954 285289093487601,3423625845396435,41084450500377465,493019048181297771
%N A017954 Expansion of 1/((1-3x)(1-6x)(1-12x)).
%K A017954 nonn
%O A017954 0,2
%A A017954 njas
%I A055434
%S A055434 1,21,317,4169,49689,532509,5260181,48218513,415055025,3375505573,
%T A055434 26107328109,193280122713,1374386800585,9405092131245,62077194367429,
%U A055434 396122100447649,2449318034512737,14705097001902901,85877415063465373
%N A055434 Points in Z^n of norm <= 10.
%K A055434 nonn
%O A055434 0,2
%A A055434 dww
%I A016315
%S A016315 1,21,319,4305,55015,683697,8369047,101581473,1227048295,
%T A016315 14781074385,177768357559,2135988547329,25651240368391,
%U A016315 307950529031985,3696355860679255,44362916914251873,532401529073793703
%N A016315 Expansion of 1/((1-2x)(1-7x)(1-12x)).
%K A016315 nonn
%O A016315 0,2
%A A016315 njas
%I A057610
%S A057610 1,21,321,4321
%N A057610 Smallest lucky number containing leading sequence of n descending numbers.
%Y A057610 Cf. A000959, A053547.
%K A057610 more,nonn
%O A057610 0,2
%A A057610 Naohiro Nomoto (n_nomoto@yabumi.com), Oct 09 2000
%I A057138
%S A057138 0,1,21,321,4321,54321,654321,7654321,87654321,987654321,987654321,
%T A057138 10987654321,210987654321,3210987654321,43210987654321,543210987654321,
%U A057138 6543210987654321,76543210987654321,876543210987654321
%N A057138 Concatenate next digit at left hand end.
%F A057138 a(n) =a(n-1)+10^(n-1)*n-10^n*floor[n/10] =A057139(n) mod 10^n.
%Y A057138 Alternative progression for n >= 10 compared with A000422 and A014925. Cf. A057137 for reverse.
%K A057138 base,easy,nonn
%O A057138 0,3
%A A057138 Henry Bottomley (se16@btinternet.com), Aug 12 2000
%I A014925
%S A014925 1,21,321,4321,54321,654321,7654321,87654321,987654321,10987654321,
%T A014925 120987654321,1320987654321,14320987654321,154320987654321,
%U A014925 1654320987654321,17654320987654321,187654320987654321
%N A014925 Number of zeros in numbers 1 to 111...1 (n+1 digits)
%F A014925 a(1)=1, a(n)=n*10^(n-1)+a(n-1). G.f.: x/((1-x)(1-10x)^2).
%F A014925 a(n) = 1/90*10^(n+1)*(n+1)-1/81*10^(n+1)+1/81.
%Y A014925 Cf. A033713.
%K A014925 nonn,easy
%O A014925 1,2
%A A014925 Olivier Gerard (ogerard@ext.jussieu.fr)
%E A014925 Better description from Stephen G. Penrice (spenrice@ets.org), Oct 03 2000
%I A000422
%S A000422 1,21,321,4321,54321,654321,7654321,87654321,987654321,10987654321,
%T A000422 1110987654321,121110987654321,13121110987654321,1413121110987654321,
%U A000422 151413121110987654321,16151413121110987654321
%N A000422 Smarandache reverse sequence.
%D A000422 F. Smarandache, "Properties of the Numbers", University of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ
%H A000422 M. L. Perez et al., eds., Smarandache Notions Journal
%H A000422 R. W. Stephan, Factors and primes in two Smarandache sequences
%H A000422 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A000422 a(n+1) = (n+1)*10^len(a(n)) + a(n), where len(k) = number of digits in k.
%K A000422 nonn,base
%O A000422 1,2
%A A000422 R. Muller
%I A060554
%S A060554 1,21,321,4321,54321,654321,7654321,87654321,987654321,98765432110,
%T A060554 9876543211110,987654321211110,98765432131211110,9876543214131211110,
%U A060554 987654321514131211110,98765432161514131211110
%N A060554 String together the first n numbers in an order which maximizes the result.
%C A060554 If n is a power of 10 then put it at the end, if it is a repdigit then put it before floor[n/10], otherwise put it before (n-1).
%Y A060554 Cf. A000422, A060555.
%K A060554 base,easy,nonn
%O A060554 1,2
%A A060554 Henry Bottomley (se16@btinternet.com), Apr 02 2001
%I A036737
%S A036737 1,1,1,21,321,4431,63429,969087,15382545,249244338,4107163191,68716945920,
%T A036737 1164359443533,19935492104010,344346982288071,5993536584212436,
%U A036737 105018405221562585,1850933501297074929,32792012556842728050,583652338245954384960
%N A036737 G.f. satisfies A(x) = 1 + x*cycle_index(G,A(x)) where G = cyclic group of order 41 generated by (1,2,...,41).
%Y A036737 Cf. A036717-A036726.
%K A036737 nonn
%O A036737 0,4
%A A036737 njas
%I A016262
%S A016262 1,21,322,4362,55363,675423,8027524,93683604,1078947205,
%T A016262 12304267305,139269572806,1567268992926,17557692150727,
%U A016262 195994212714867,2181672731375368,24230027568735528,268614950968549129
%N A016262 Expansion of 1/((1-x)(1-9x)(1-11x)).
%K A016262 nonn
%O A016262 0,2
%A A016262 njas
%I A001233 M5114 N2216
%S A001233 1,21,322,4536,63273,902055,13339535,206070150,3336118786,
%T A001233 56663366760,1009672107080,18861567058880,369012649234384,
%U A001233 7551527592063024
%N A001233 Stirling numbers of first kind.
%D A001233 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
%D A001233 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
%F A001233 E.g.f.: (-log(1-x))^6 or (1-x)^-1 * (-log(1-x))^5.
%F A001233 a(n) is coefficient of x^(n+6) in (-log(1-x))^6, multiplied by (n+6)!/6!.
%e A001233 (-log(1-x))^6 = x^6 + 3*x^7 + 23/4*x^8 + 9*x^9 + ...
%Y A001233 Cf. A000254, A000399, A000454, A000482, A008275 (Stirling1 triangle).
%K A001233 nonn
%O A001233 6,2
%A A001233 njas
%I A016260
%S A016260 1,21,325,4485,58501,739461,9173125,112474245,1368864901,
%T A016260 16579770501,200184379525,2412029622405,29022892013701,
%U A016260 348902996523141,4191862297147525,50342558276728965,604432385008416901
%N A016260 Expansion of 1/((1-x)(1-8x)(1-12x)).
%K A016260 nonn
%O A016260 0,2
%A A016260 njas
%I A011810
%S A011810 0,0,1,21,330,4931,76064,1257345,22707559,452800399,10023029299,
%T A011810 246756095614,6752175162108,204875755372586,6868905893537535,
%U A011810 253432530143048034,10246233216680784186,452038162651568624259
%N A011810 M-sequences from multicomplexes on 4 variables with all monomials of degree 2 but none of degree larger than n.
%K A011810 nonn
%O A011810 1,4
%A A011810 Svante Linusson (linusson@math.kth.se)
%I A016195
%S A016195 0,1,21,331,4641,61051,771561,9487171,114358881,1357947691,
%T A016195 15937424601,185311670611,2138428376721,24522712143931,
%U A016195 279749833583241,3177248169415651,35949729863572161,405447028499293771
%N A016195 11^n - 10^n.
%F A016195 G.f.: x/((1-10x)(1-11x)).
%K A016195 nonn
%O A016195 0,3
%A A016195 njas
%I A016191
%S A016191 1,21,333,4725,63261,818181,10349613,128978325,1590786621,
%T A016191 19476859941,237209103693,2877890303925,34817113183581,
%U A016191 420347224031301,5067043480830573,61010412902061525,733977975013590141
%N A016191 Expansion of 1/((1-9x)(1-12x)).
%K A016191 nonn
%O A016191 0,2
%A A016191 njas
%I A051525
%S A051525 0,0,1,21,335,5000,74524,1139292,18083484,299705400,5198985576,
%T A051525 94461323616,1797180658272,35776357096896,744402741205824,
%U A051525 16169795109262080,366214212167489280,8636605663418933760
%N A051525 Third unsigned column of triangle A051338.
%D A051525 Mitrinovic, D. S. and Mitrinovic, R. S. see ref. given for triangle A051338.
%F A051525 a(n) = A051338(n,2)*(-1)^n; e.g.f.: (ln(1-x))^2/(2*(1-x)^6).
%Y A051525 A001725 (m=0), A051524 (m=1) unsigned columns.
%K A051525 easy,nonn
%O A051525 0,4
%A A051525 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A036224
%S A036224 1,21,336,4536,54432,598752,6158592,60046272,560431872,5043886848,
%T A036224 44019376128,374164697088,3108445175808,25311625003008,202493000024064,
%U A036224 1594632375189504,12381851383824384,94927527275986944
%N A036224 Expansion of (-1+1/(1-6*x)^6)/(36*x); related to A036084.
%F A036224 a(n) = 6^(n-1)*binomial(n+6,5); G.f. (-1+(1-6*x)^(-6))/(x*6^2).
%Y A036224 A036084, A036083.
%K A036224 easy,nonn
%O A036224 0,2
%A A036224 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A062260
%S A062260 1,21,336,5040,75600,1164240,18627840,311351040,5448643200,99891792000,
%T A062260 1917922406400,38532804710400,809188898918400,17739910476288000,
%U A062260 405483668029440000,9650511299100672000
%N A062260 Third (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).
%H A062260 Index entries for sequences related to Laguerre polynomials
%F A062260 E.g.f.: (1+12*x+15*x^2)/(1-x)^9.
%F A062260 a(n)=A062140(n+2,2) = (n+2)!*binomial(n+6,6)/2!.
%Y A062260 A001720, A062199.
%K A062260 nonn,easy
%O A062260 0,2
%A A062260 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Jun 19 2001
%I A020311
%S A020311 21,341,781,1591,2201,3097,4859,7141,7613,8299,13333,13817,14689,17059,
%T A020311 17711,23441,24211,24641,24727,32167,32581,37291,45991,46969,49321,58969,
%U A020311 66061,67069,75851,77449,79003,83381,85457,94831,102311,104777,104833
%N A020311 Strong pseudoprimes to base 85.
%H A020311 Index entries for sequences related to pseudoprimes
%K A020311 nonn
%O A020311 1,1
%A A020311 dww
%I A006105 M5115
%S A006105 1,21,357,5797,93093,1490853,23859109,381767589,6108368805,97734250405,
%T A006105 1563749404581,25019996065701,400319959420837,6405119440211877,102481911401303973
%N A006105 Gaussian binomial coefficient [ n,2 ] for q=4.
%D A006105 J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
%D A006105 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p, 99.
%D A006105 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%K A006105 nonn
%O A006105 2,2
%A A006105 njas
%I A051564
%S A051564 0,1,21,362,6026,101524,1763100,31813200,598482000,11752855200,
%T A051564 240947474400,5154170774400,114942011990400,2669517204076800,
%U A051564 64496340380102400,1619153396908185600,42188624389562112000
%N A051564 Second unsigned column of triangle A051523.
%D A051564 Mitrinovic, D. S. and Mitrinovic, R. S. see ref. given for triangle A051523.
%F A051564 a(n) = A051523(n,2)*(-1)^(n-1); e.g.f.: -ln(1-x)/(1-x)^10.
%Y A051564 Cf. A049398 (first unsigned column).
%K A051564 easy,nonn
%O A051564 0,3
%A A051564 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A036904
%S A036904 21,372,8092,102128,1061613,12108841
%N A036904 Scan decimal expansion of e until all n-digit strings have been seen; a(n) is number of digits that must be scanned.
%H A036904 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A036904 Cf. A032510, A036900-A036906.
%K A036904 nonn
%O A036904 1,1
%A A036904 Michael Kleber (kleber@math.mit.edu)
%I A004370
%S A004370 1,21,378,6545,111930,1906884,32468436,553270671,9440350920,
%T A004370 161322559475,2761025887620,47325339895743,812325612855848,
%U A004370 13961746143269400,240260199935164200,4139207762053520646
%N A004370 Binomial coefficient C(7n,n-2).
%D A004370 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.
%K A004370 nonn,easy
%O A004370 2,2
%A A004370 njas
%I A001881 M5116 N2217
%S A001881 1,21,378,6930,135135,2837835,64324260,1571349780,41247931725,1159525191825,
%T A001881 34785755754750,1109981842719750,37554385678684875,1343291487737574375
%N A001881 Coefficients of Bessel polynomials y_n (x).
%D A001881 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A001881 Index entries for sequences related to Bessel functions or polynomials
%Y A001881 See A001518.
%K A001881 nonn,easy
%O A001881 5,2
%A A001881 njas, Simon Plouffe (plouffe@math.uqam.ca)
%I A015677
%S A015677 1,21,399,6783,102249,1370565,16991415,216411615,3146775345,
%T A015677 49819225365,767467173375,11808542505375,208290421179225,
%U A015677 4084751136190725,76593243172220775,1426967042034909375
%N A015677 Expansion of theta_3^(41/2).
%D A015677 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%K A015677 nonn
%O A015677 0,2
%A A015677 njas
%I A014903
%S A014903 1,21,402,7642,145203,2758863,52418404,995949684,18923044005,
%T A014903 359537836105,6831218886006,129793158834126,2466070017848407,
%U A014903 46855330339119747,890251276443275208,16914774252422228968
%N A014903 a(1)=1, a(n)=19*a(n-1)+n.
%K A014903 nonn
%O A014903 1,2
%A A014903 njas, Olivier Gerard (ogerard@ext.jussieu.fr)
%I A020534
%S A020534 21,408,23184,2298912,274767936,34561392768,4404491583744,
%T A020534 563156132823552,72064191275467776,9223583144429488128,
%U A020534 1180598376127589781504,151115943624696659976192
%N A020534 8th Fibonacci polynomial evaluated at powers of 2.
%p A020534 with(combinat,fibonacci):seq(fibonacci(8,2**i),i=0..24);
%K A020534 nonn
%O A020534 0,1
%A A020534 Simon Plouffe (plouffe@math.uqam.ca)
%I A064108
%S A064108 1,21,421,8421,168421,3368421,67368421,1347368421,26947368421,
%T A064108 538947368421,10778947368421,215578947368421,4311578947368421,
%U A064108 86231578947368421,1724631578947368421,34492631578947368421
%N A064108 (20^n-1)/19.
%o A064108 (PARI.2.0.17) for(n=1,23,print((20^n-1)/19))
%K A064108 easy,nonn
%O A064108 1,2
%A A064108 Jason Earls (jcearls@kskc.net), Sep 17 2001
%I A009965
%S A009965 1,21,441,9261,194481,4084101,85766121,1801088541,37822859361,
%T A009965 794280046581,16679880978201,350277500542221,7355827511386641,
%U A009965 154472377739119461,3243919932521508681,68122318582951682301
%N A009965 Powers of 21.
%F A009965 For A009966-A009992 we have g.f.: 1/(1-qx), e.g.f.: exp(qx), with q = 21,22,...,48. - Dan Fux (danfux@my-deja.com), Apr 07 2001
%K A009965 nonn
%O A009965 0,2
%A A009965 njas
%I A041842
%S A041842 21,442,18585,390727,16429119,345402226,14523322611,305335177057,
%T A041842 12838600759005,269915951116162,11349308547637809,238605395451510151,
%U A041842 10032775917511064151,210926899663183857322,8868962561771233071675
%N A041842 Numerators of continued fraction convergents to sqrt(443).
%Y A041842 Cf. A041843.
%K A041842 nonn,cofr,easy
%O A041842 0,1
%A A041842 njas
%I A025603
%S A025603 1,21,465,10565,241697,5539893,127041105,2913686981,66827609633,
%T A025603 1532754884725,35155272163473,806321934125125,18493816732267425,
%U A025603 424174583966543669,9728877611505065297
%N A025603 n-move queen paths on 8x8 board from given corner to any square.
%K A025603 nonn
%O A025603 0,2
%A A025603 dww
%I A006298 M5117
%S A006298 21,483,6468,66066,570570,4390386,31039008,205633428,1293938646,
%T A006298 7808250450,45510945480,257611421340,1422156202740,7683009544980,
%U A006298 40729207226400,212347275857640,1090848505817070,5530195966465170
%N A006298 Rooted genus-2 maps with n edges.
%D A006298 T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.
%D A006298 T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
%H A006298 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A006298 a(n+1) = ((5n+3)(4n+2)a(n))/((5n-2)(n-3))
%Y A006298 Cf. A035309.
%K A006298 nonn,easy
%O A006298 4,1
%A A006298 njas
%E A006298 More terms from dww
%I A015695
%S A015695 1,21,483,12075,325521,9396765,288749475,9398322675,322652227905,
%T A015695 11641352627205,440043353374275,17378800976950875,715361536295492625,
%U A015695 30625157300600587725,1360947531581394631875,62670194480910433291875
%V A015695 1,-21,483,-12075,325521,-9396765,288749475,-9398322675,322652227905,
%W A015695 -11641352627205,440043353374275,-17378800976950875,715361536295492625,
%X A015695 -30625157300600587725,1360947531581394631875,-62670194480910433291875
%N A015695 Expansion of theta_3^(-41/2).
%D A015695 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%K A015695 sign,done
%O A015695 0,2
%A A015695 njas
%I A006299 M5118
%S A006299 21,483,15018,258972,5554188,85421118,1558792200,22555934280,375708427812,
%T A006299 5235847653036,82234427131416
%N A006299 Rooted genus-2 maps with n edges.
%D A006299 T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.
%D A006299 T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
%K A006299 nonn
%O A006299 4,1
%A A006299 njas
%I A065921
%S A065921 0,1,21,501,14455,496770,19911486,913839031,47303189361,2727741976785,
%T A065921 173455231572865,12060173714421756,910301022642409476,74134150415555474881,
%U A065921 6479678618270868170265,605042444997867941987385,60110944381660549838273911
%N A065921 Bessel polynomial {y_n}'(3).
%D A065921 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A065921 Index entries for sequences related to Bessel functions or polynomials
%Y A065921 Cf. A001514, A065920, A065922.
%K A065921 nonn
%O A065921 0,3
%A A065921 njas, Dec 08 2001
%I A015255
%S A015255 1,21,546,13546,339171,8476671,211929796,5298179796,132454820421,
%T A015255 3311368882921,82784230211046,2069605714586046,51740143068101671,
%U A015255 1293503575685289171,32337589397218492296,808439734905030992296
%N A015255 Gaussian binomial coefficient [ n,2 ] for q=-5.
%D A015255 J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
%D A015255 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p, 99.
%D A015255 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%K A015255 nonn,easy,huge
%O A015255 2,2
%A A015255 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A034789
%S A034789 1,21,546,15561,466830,14471730,458960580,14801478705,483514971030,
%T A034789 15955994043990,530899438190940,17785131179396490,599222112044281740,
%U A034789 20287948650642110340,689790254121831751560,23539092421907508521985
%N A034789 Related to sextic factorial numbers A008542.
%C A034789 Convolution of A004993(n-1) with A025751(n), n >= 1.
%F A034789 a(n) = 6^(n-1)*A008542(n)/n!, A008542(n)=(6*n-5)(!^6):= product(6*j-5,j=1..n); G.f. (-1+(1-36*x)^(-1/6))/6.
%Y A034789 Cf. A008542, A034687.
%K A034789 easy,nonn
%O A034789 1,2
%A A034789 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de)
%I A009091
%S A009091 1,1,21,597,26537,1834249,180938109,23781199197,3989919982673,
%T A009091 830092069266577,209479116176732901,62986349633232396261,
%U A009091 22231865269167169440761,9096170536950779556167833
%V A009091 1,-1,21,-597,26537,-1834249,180938109,-23781199197,3989919982673,
%W A009091 -830092069266577,209479116176732901,-62986349633232396261,
%X A009091 22231865269167169440761,-9096170536950779556167833
%N A009091 Expansion of cos(tanh(x).cos(x)).
%t A009091 Cos[ Tanh[ x ]*Cos[ x ] ] (* Even Part *)
%K A009091 sign,done
%O A009091 0,3
%A A009091 R. H. Hardin (rhh@research.bell-labs.com)
%E A009091 Extended with signs 03/97 by Olivier Gerard.
%I A025752
%S A025752 1,1,21,637,22295,842751,33429123,1370594043,57564949806,
%T A025752 2462500630590,106872527367606,4692675519868518,208041948047504298,
%U A025752 9297874755046153626,418404363977076913170,18939770876029014936162
%N A025752 7th order Patalan numbers (generalization of Catalan numbers).
%H A025752 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A025752 G.f.: (8-(1-49*x)^(1/7))/7.
%F A025752 a(n)= 7^(n-1)*6*A034833(n-1)/n!, n >= 2, 6*A034833(n-1)= (7*n-8)(!^7):= product(7*j-8,j=2..n) - Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de).
%K A025752 nonn,easy
%O A025752 0,3
%A A025752 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A012501
%S A012501 1,1,21,645,34153,2966505,380114493,67244074989,15717842398417,
%T A012501 4694489063253585,1744673430747688805,789702534539438543573,
%U A012501 427739642844597474379833,273184203052102765688128825
%V A012501 1,-1,21,-645,34153,-2966505,380114493,-67244074989,15717842398417,
%W A012501 -4694489063253585,1744673430747688805,-789702534539438543573,
%X A012501 427739642844597474379833,-273184203052102765688128825
%N A012501 cos(cos(x)*arctan(x))=1-1/2!*x^2+21/4!*x^4-645/6!*x^6+34153/8!*x^8...
%K A012501 sign,done
%O A012501 0,3
%A A012501 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A027408
%S A027408 1,21,651,25151,1141791,58999227,3398446177,215000489757,
%T A027408 14773746912897,1093126139653877,86488314164428659,
%U A027408 7275617735214635079,647633785826931486079,60753264284081214177339
%N A027408 Labeled servers of dimension 21.
%D A027408 R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
%F A027408 E.g.f.: Exp[ Sum[ ((1+x)^i-1)/i,{i, 1, 21} ] ]
%K A027408 nonn
%O A027408 0,2
%A A027408 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A020246
%S A020246 21,671,889,1891,2059,2761,5461,7999,13051,15311,16441,21667,25681,34861,
%T A020246 37901,38989,42127,49771,50737,54811,64681,68251,78961,85591,88831,92509,
%U A020246 93031,96049,97921,105001,109061,111361,114841,123713,143009,145351
%N A020246 Strong pseudoprimes to base 20.
%H A020246 Index entries for sequences related to pseudoprimes
%K A020246 nonn
%O A020246 1,1
%A A020246 dww
%I A006934 M5119
%S A006934 1,1,21,671,180323,20898423,7426362705,1874409465055
%N A006934 A series for pi.
%D A006934 Y. L. Luke, The Special Functions and their Approximation, Vol. 1, Academic Press, NY, 1969, see p. 36.
%K A006934 nonn
%O A006934 0,3
%A A006934 sp, njas
%I A056565
%S A056565 1,21,714,19635,582505,16776144,488605194,14169550626,411591708660,
%T A056565 11948265189630,346934172869802,10072785423545712,292460526776698763,
%U A056565 8491396839675395415,246543315138161480670,7158243695757340957617
%N A056565 Fibonomial coefficients.
%F A056565 a(n)= A010048(n+7,7)=: fibonomial(n+7,7).
%F A056565 G.f. 1/p(8,n) with p(8,n)= 1-21*x-273*x^2+1092*x^3+1820*x^4-1092*x^5-273*x^6+21*x^7+x^8 = (1+x-x^2)*(1-4*x-x^2)*(1+11*x-x^2)*(1-29*x-x^2) (n=8 row polynomial of signed Fibonomial triangle A055870; see this A-number for Knuth and Riordan refs.).
%F A056565 Recursion: a(n)=29*a(n-1)+a(n-2)+((-1)^n)*A001657(n), n >= 2, a(0)=1, a(1)=21.
%Y A056565 Cf. A010048, A000045, A001654-8, A001076, A049666 (signed), A049667.
%K A056565 nonn,easy
%O A056565 0,2
%A A056565 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de) Jul 10 2000
%I A009167
%S A009167 1,1,21,717,38857,3158329,363525789,56250039237,11210469917713,
%T A009167 2783865475754737,840008000833503141,302092032859131150141,
%U A009167 127513257396264637784281,62381091849825754078253353
%N A009167 Expansion of cosh(tan(x)/cos(x)).
%t A009167 Cosh[ Tan[ x ]/Cos[ x ] ] (* Even Part *)
%K A009167 nonn
%O A009167 0,3
%A A009167 R. H. Hardin (rhh@research.bell-labs.com)
%E A009167 Extended and signs tested 03/97 by Olivier Gerard.
%I A012479
%S A012479 1,1,21,717,46025,4817209,734055069,154121194437,42674286160913,
%T A012479 15062120873888497,6601538801733818277,3517682024725193957181,
%U A012479 2239550167783452398617817,1678950113256157460626211113
%V A012479 1,-1,21,-717,46025,-4817209,734055069,-154121194437,42674286160913,
%W A012479 -15062120873888497,6601538801733818277,-3517682024725193957181,
%X A012479 2239550167783452398617817,-1678950113256157460626211113
%N A012479 sech(cos(x)*sin(x))=1-1/2!*x^2+21/4!*x^4-717/6!*x^6+46025/8!*x^8...
%K A012479 sign,done
%O A012479 0,3
%A A012479 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A062755
%S A062755 1,21,757,69905,9768751,2214363531,678223896393,282578800148737,
%T A062755 150102261281924281,100097761867442455851,81402749387125072783933,
%U A062755 79516409977044969123349715,91733330193268919533506208263
%N A062755 sigma_n(n^2): sum of n-th powers of divisors of n^2.
%o A062755 (PARI.2.0.17) for(n=1,22,print(sigma(n^2,n)))
%K A062755 huge,nonn
%O A062755 1,2
%A A062755 Jason Earls (jcearls@kskc.net), Jul 15 2001
%I A012850
%S A012850 1,1,21,765,46473,4371225,594635613,110691178325,27018807872657,
%T A012850 8370279766396593,3208469378545620005,1491108362472114151469,
%U A012850 826269720877603294674201,538334287550086657620232905
%N A012850 cosh(sec(x)*arctanh(x))=1+1/2!*x^2+21/4!*x^4+765/6!*x^6+46473/8!*x^8...
%K A012850 nonn
%O A012850 0,3
%A A012850 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A012645
%S A012645 1,1,21,765,53193,6008025,993328413,226691205013,68261145008657,
%T A012645 26216583047151921,12507068079360896165,7255577391165958142893,
%U A012645 5029681350011635244424921,4106028806747521109741788425
%V A012645 1,-1,21,-765,53193,-6008025,993328413,-226691205013,68261145008657,
%W A012645 -26216583047151921,12507068079360896165,-7255577391165958142893,
%X A012645 5029681350011635244424921,-4106028806747521109741788425
%N A012645 sech(arcsinh(x)*cos(x))=1-1/2!*x^2+21/4!*x^4-765/6!*x^6+53193/8!*x^8...
%K A012645 sign,done
%O A012645 0,3
%A A012645 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A028469
%S A028469 21,781,31529,1292697,53175517,2188978117,90124167441,3710708201969,
%T A028469 152783289861989,6290652543875133,259009513044645817,
%U A028469 10664383939345916681,439092316687230373293,18079062471131097321077
%N A028469 Number of perfect matchings in graph P_{7} X P_{2n}.
%D A028469 P.H. Lundow, "Computation of matching polynomials and the number of 1-factors in polygraphs", Research report, No 12, 1996, Department of Math., Umea University, Sweden.
%H A028469 P.-H. Lundow, Enumeration of matchings in polygraphs, 1998.
%K A028469 nonn
%O A028469 1,1
%A A028469 Per-Hakan Lundow (per-hakan.lundow@math.umu.se)
%I A012819
%S A012819 1,1,21,837,60585,7048489,1201350909,282008557677,87244033155153,
%T A012819 34402407033544657,16843665951415524837,10025536600336492439061,
%U A012819 7129473645057562412172537,5969969671887834658217008633
%N A012819 sec(sec(x)*sinh(x))=1+1/2!*x^2+21/4!*x^4+837/6!*x^6+60585/8!*x^8...
%K A012819 nonn
%O A012819 0,3
%A A012819 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A041843
%S A041843 1,21,883,18564,780571,16410555,690023881,14506912056,609980330233,
%T A041843 12824093846949,539221921902091,11336484453790860,476671568981118211,
%U A041843 10021439433057273291,421377127757386596433,8858941122338175798384
%N A041843 Denominators of continued fraction convergents to sqrt(443).
%Y A041843 Cf. A041842.
%K A041843 nonn,cofr,easy
%O A041843 0,2
%A A041843 njas
%I A041840
%S A041840 21,883,37107,1559377,65530941,2753858899,115727604699,
%T A041840 4863313256257,204374884367493,8588608456690963,360925930065387939,
%U A041840 15167477671202984401,637394988120590732781,26785756978736013761203
%N A041840 Numerators of continued fraction convergents to sqrt(442).
%Y A041840 Cf. A041841.
%K A041840 nonn,cofr,easy
%O A041840 0,1
%A A041840 njas
%I A012793
%S A012793 1,1,21,885,67753,8319945,1497879933,371676224829,121584750716497,
%T A012793 50703653565909009,26255891609845966565,16529283187035337691269,
%U A012793 12432784542297124704745209,11011634249882125250110463705
%N A012793 sec(sec(x)*arcsin(x))=1+1/2!*x^2+21/4!*x^4+885/6!*x^6+67753/8!*x^8...
%K A012793 nonn
%O A012793 0,3
%A A012793 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A015305
%S A015305 1,21,903,25585,875007,27125217,882215391,28005209505,899790907743,
%T A015305 28735427761313,920460637644639,29439916001972385,942314556807454559,
%U A015305 30150270336284213409,964869381941043396447,30874848551033891160225
%V A015305 1,-21,903,-25585,875007,-27125217,882215391,-28005209505,899790907743,
%W A015305 -28735427761313,920460637644639,-29439916001972385,942314556807454559,
%X A015305 -30150270336284213409,964869381941043396447,-30874848551033891160225
%N A015305 Gaussian binomial coefficient [ n,5 ] for q=-2.
%D A015305 J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
%D A015305 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p, 99.
%D A015305 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%K A015305 sign,done,easy,huge
%O A015305 5,2
%A A015305 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A006301 M5120
%S A006301 0,0,0,0,21,966,27954,650076,13271982,248371380,4366441128,
%T A006301 73231116024,1183803697278,18579191525700,284601154513452
%N A006301 Rooted genus-2 maps with n edges.
%D A006301 T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.
%D A006301 T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
%D A006301 E. A. Bender and E. R. Canfield, The number of rooted maps on an orientable surface, J. Combin. Theory, B 53 (1991), 293-299.
%K A006301 nonn
%O A006301 0,5
%A A006301 njas. Simon Plouffe (plouffe@math.uqam.ca)
%I A004704
%S A004704 1,21,973,67473,6238309,720964881,99986786773,16177741934193,
%T A004704 2991473373828709,622307309978695761,143840821212045590773,
%U A004704 36572284571798550251313,10144031468802588684994309,3048113900510603294243693841
%N A004704 Expansion of 1/(7-exp(x)-exp(2*x)-exp(3*x)-exp(4*x)-exp(5*x)-exp(6*x)).
%K A004704 nonn
%O A004704 0,2
%A A004704 njas
%I A012153
%S A012153 1,1,21,1053,96905,14216409,3045819741,897606889013,348348521840273,
%T A012153 172224469448231985,105685328498514238501,78822199814116989651661,
%U A012153 70223663543654715642543129,73659278933239564262294052361
%N A012153 sec(tan(tan(x)))=1+1/2!*x^2+21/4!*x^4+1053/6!*x^6+96905/8!*x^8...
%K A012153 nonn
%O A012153 0,3
%A A012153 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A012183
%S A012183 1,1,21,1101,107209,16711481,3810146205,1195529025861,
%T A012183 494050576455185,260086639760540401,169928807728242392485,
%U A012183 134924876918419856397501,127963083862513681084495321
%N A012183 sec(tan(arctanh(x)))=1+1/2!*x^2+21/4!*x^4+1101/6!*x^6+107209/8!*x^8...
%K A012183 nonn
%O A012183 0,3
%A A012183 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A012230
%S A012230 1,1,21,1101,109001,17491001,4139085981,1355308485701,
%T A012230 586612833746961,324261948714025201,222851423174495787941,
%U A012230 186365710421614910029501,186332939048243649892744921
%V A012230 1,-1,21,-1101,109001,-17491001,4139085981,-1355308485701,
%W A012230 586612833746961,-324261948714025201,222851423174495787941,
%X A012230 -186365710421614910029501,186332939048243649892744921
%N A012230 sech(arctan(tanh(x)))=1-1/2!*x^2+21/4!*x^4-1101/6!*x^6+109001/8!*x^8...
%K A012230 sign,done
%O A012230 0,3
%A A012230 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A012211
%S A012211 1,1,21,1149,119305,20082841,4984310493,1711500674389,
%T A012211 776816116717457,450301381427975089,324553968807761095973,
%U A012211 284661632752853556677805,298518404617173649259722905
%V A012211 1,-1,21,-1149,119305,-20082841,4984310493,-1711500674389,
%W A012211 776816116717457,-450301381427975089,324553968807761095973,
%X A012211 -284661632752853556677805,298518404617173649259722905
%N A012211 sech(arctan(atan(x)))=1-1/2!*x^2+21/4!*x^4-1149/6!*x^6+119305/8!*x^8...
%K A012211 sign,done
%O A012211 0,3
%A A012211 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A036059
%S A036059 1,1,21,1221,3231,233231,533221,15534221,3514334231,3534533241,
%T A036059 3544832231,183544733221,28172544634231,2827162554535241,
%U A036059 2827265554337241,2837267544338231,3847264544637221,3847362564636221
%N A036059 The summarize Fibonacci sequence: summarize the previous two terms!.
%C A036059 After the 23rd term the sequence goes into a cycle of 16 terms.
%Y A036059 Cf. A036058, A036066.
%K A036059 nonn,base,nice,easy
%O A036059 0,3
%A A036059 Floor van Lamoen (f.v.lamoen@wxs.nl)
%I A036519
%S A036519 21,1225,23220,212226,22121226,2202322528,21222223210,202222242820,
%T A036519 22122952222920,229222922294226,2221228522222920
%N A036519 Smallest triangle containing exactly n 2's.
%K A036519 nonn,base
%O A036519 1,1
%A A036519 dww
%I A035319
%S A035319 1,1,21,1485,225225
%N A035319 Right-hand diagonal of A035309.
%K A035319 nonn
%O A035319 0,3
%A A035319 njas
%I A007593 M5121
%S A007593 21,2133,19521,176661,129127041
%N A007593 2-hyperperfect numbers: n=2(sigma(n)-n-1)+1.
%D A007593 D. Minoli, Issues in non-linear hyperperfect numbers, Math. Comp., 34 (1980), 639-645.
%D A007593 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.
%D A007593 R. K. Guy, Unsolved Problems in Number Theory, B2.
%H A007593 J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3
%H A007593 E. W. Weisstein, Link to a section of The World of Mathematics.
%K A007593 nonn,hard
%O A007593 1,1
%A A007593 njas, dww
%E A007593 328256967373616371221, 67585198634817522935331173030319681, 443426488243037769923934299701036035201 are also in the sequence - Jud McCranie (jud.mccranie@mindspring.com), Dec 16 1999
%I A033510
%S A033510 21,2356,196785,17525619,1539222016,135658637925,11945257052321,
%T A033510 1052091957273408,92657526436631289,8160498611028648795,
%U A033510 718704019165239462736,63297158846544276862187
%N A033510 Matchings in graph P_{7} X P_{n}
%D A033510 Per Hakan Lundow, "Computation of matching polynomials and the number of 1-factors in polygraphs", Research reports, No 12, 1996, Department of Mathematics, Umea University.
%H A033510 P.-H. Lundow, Enumeration of matchings in polygraphs, 1998.
%K A033510 nonn
%O A033510 0,1
%A A033510 P.H. Lundow (per-hakan.lundow@math.umu.se)
%I A018238
%S A018238 1,21,3121,41213121,5121312141213121,
%T A018238 61213121412131215121312141213121,
%U A018238 7121312141213121512131214121312161213121412131215121312141213121
%N A018238 Add 1 to leading digit and put in front.
%Y A018238 Cf. A001511.
%K A018238 nonn,base
%O A018238 1,2
%A A018238 njas, Michael Minic (minic@mtsu.edu)
%I A055416
%S A055416 1,21,4541,198765,3083569,27634481,164379601,759891589,2839094517,
%T A055416 9183188589,26107328109,67602028569,160441685209,357086356401,
%U A055416 746545031221,1487788785845,2829595966381,5188248484757,9170828884817
%N A055416 Points in Z^10 of norm <= n.
%K A055416 nonn
%O A055416 0,2
%A A055416 dww
%I A028451
%S A028451 21,6272,880163,152526845,24972353440,4161756233501,690427159718433,
%T A028451 114725843769441312,19056798015394695543,3165986817537284900809,
%U A028451 525966380704787334395776,87380576637559587656345353
%N A028451 Perfect matchings in graph P_{2} X P_{7} X P_{n}.
%D A028451 P.H. Lundow, "Computation of matching polynomials and the number of 1-factors in polygraphs", Research report, No 12, 1996, Department of Math., Umea University, Sweden.
%H A028451 P.-H. Lundow, Enumeration of matchings in polygraphs, 1998.
%K A028451 nonn
%O A028451 1,1
%A A028451 Per-Hakan Lundow (per-hakan.lundow@math.umu.se)
%I A013726
%S A013726 21,9261,4084101,1801088541,794280046581,350277500542221,
%T A013726 154472377739119461,68122318582951682301,30041942495081691894741,
%U A013726 13248496640331026125580781,5842587018385982521381124421
%N A013726 21^(2n+1).
%K A013726 nonn,easy
%O A013726 0,1
%A A013726 njas
%I A048914
%S A048914 1,21,10981,252081,132846121,3049673901,1607172358861,36894954600201,
%T A048914 19443571064652241,446355157703555781,235228321132990450741,
%U A048914 5400004661002663236321,2845792209623347408410361
%N A048914 Indices of pentagonal numbers which are also 9-gonal.
%H A048914 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A048914 Cf. A048913, A048915.
%K A048914 nonn
%O A048914 1,2
%A A048914 Eric W. Weisstein (eric@weisstein.com)
%I A046183
%S A046183 1,21,11781,203841,113123361,1957283461,1086210502741,18793835590881,
%T A046183 10429793134197921,180458407386358101,100146872588357936901,
%U A046183 1732761608929974897121,961610260163619775927681
%N A046183 Octagonal triangular numbers.
%H A046183 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A046183 Cf. A046181, A046182.
%K A046183 nonn
%O A046183 1,2
%A A046183 Eric W. Weisstein (eric@weisstein.com)
%I A001167 M5122 N2218
%S A001167 1,21,21000,101,121,1101,1121,21121,101101,101121,121121,1101121,
%T A001167 1121121,21121121,101101121,101121121,121121121,1101121121,1121121121
%N A001167 Smallest natural number requiring n words in English.
%H A001167 English names for the numbers from 0 to 1022, from G. Schildberger
%K A001167 nonn
%O A001167 1,2
%A A001167 njas
%I A013768
%S A013768 21,194481,1801088541,16679880978201,154472377739119461,
%T A013768 1430568690241985328321,13248496640331026125580781,122694327386105632949003612841,
%U A013768 1136272165922724266740722458520501,10523016528610349434285830688358359761
%N A013768 21^(3n+1).
%K A013768 nonn
%O A013768 0,1
%A A013768 njas
%I A033636
%S A033636 1,21,1025310,393143628567690,4161601591128480023923529880,
%T A033636 3025979870886810157320547152897281546399520600,
%U A033636 316069513961213168272637175554794298273860325049869267706740636689200
%N A033636 (n-1)!*product( (2*n)!-2*i-1, i=1..n-1).
%D A033636 Barry Simon, Representations of Finite and Compact Groups, AMS, p. 233.
%K A033636 nonn,easy,huge
%O A033636 1,2
%A A033636 njas
%I A013814
%S A013814 21,4084101,794280046581,154472377739119461,30041942495081691894741,
%T A013814 5842587018385982521381124421,1136272165922724266740722458520501,
%U A013814 220983347100817338120002444455525554981,42977062327514056734916195400155065458259861
%N A013814 21^(4n+1).
%K A013814 nonn
%O A013814 0,1
%A A013814 njas
%I A013898
%S A013898 21,85766121,350277500542221,1430568690241985328321,5842587018385982521381124421,
%T A013898 23861715484377209601555171628930521,97453656071460446110921078004886769746621,
%U A013898 398010574215107679422058885600836061208944572721,1625515384162495488635310116741260158419511738394408821
%N A013898 21^(5n+1).
%K A013898 nonn
%O A013898 0,1
%A A013898 njas
%I A062012
%S A062012 21,1066338805156287287067,1124161332414632881704,
%T A062012 2305867155177711644802,2306166776784312535170,5744341611556736174883
%N A062012 Digit strings N for which there is some base b so that (N in base b) = 1/k (N in base 10).
%e A062012 21 in base 3 is 21/3
%K A062012 base,fini,full,huge,nice,nonn
%O A062012 0,1
%A A062012 Erich Friedman (efriedma@stetson.edu), Jun 27 2001
%I A023922
%S A023922 1,0,0,0,0,22,0,0,0,110,0,110,330,0,660,924,990,0,0,0,2662,0,1980,
%T A023922 4840,0,6534,7260,9460,0,0,0,15840,0,10230,21780,0,27830,32670,
%U A023922 33660,0,0
%N A023922 Theta series of A*_10 lattice.
%D A023922 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.
%H A023922 G. Nebe and N. J. A. Sloane, Home page for this lattice
%K A023922 nonn
%O A023922 0,6
%A A023922 njas
%I A022064
%S A022064 1,0,0,0,22,0,0,0,220,0,0,2048,1320,0,0,0,5302,0,0,22528,
%T A022064 15224,0,0,0,33528,0,0,112640,63360,0,0,0,116380,0,0,360448,
%U A022064 209550,0,0,0,339064,0,0,901120,491768,0,0,0,719400,0,0
%N A022064 Theta series of D*_11 lattice.
%D A022064 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.
%H A022064 G. Nebe and N. J. A. Sloane, Home page for this lattice
%K A022064 nonn
%O A022064 0,5
%A A022064 njas
%I A040488
%S A040488 22,1,1,1,1,6,1,14,4,1,21,1,4,14,1,6,1,1,1,1,44,1,1,1,1,6,1,14,
%T A040488 4,1,21,1,4,14,1,6,1,1,1,1,44,1,1,1,1,6,1,14,4,1,21,1,4,14,1,6,
%U A040488 1,1,1,1,44,1,1,1,1,6,1,14,4,1,21,1,4,14,1,6,1,1,1,1,44,1,1,1,1
%N A040488 Continued fraction for sqrt(511).
%H A040488 Index entries for continued fractions for constants
%p A040488 with(numtheory): Digits:=300: convert(evalf(sqrt(511)),confrac);
%K A040488 nonn,cofr,easy
%O A040488 0,1
%A A040488 njas
%I A040489
%S A040489 22,1,1,1,2,6,11,6,2,1,1,1,44,1,1,1,2,6,11,6,2,1,1,1,44,1,1,1,2,
%T A040489 6,11,6,2,1,1,1,44,1,1,1,2,6,11,6,2,1,1,1,44,1,1,1,2,6,11,6,2,1,
%U A040489 1,1,44,1,1,1,2,6,11,6,2,1,1,1,44,1,1,1,2,6,11,6,2,1,1,1,44,1,1
%N A040489 Continued fraction for sqrt(512).
%H A040489 Index entries for continued fractions for constants
%p A040489 with(numtheory): Digits:=300: convert(evalf(sqrt(512)),confrac);
%K A040489 nonn,cofr,easy
%O A040489 0,1
%A A040489 njas
%I A040490
%S A040490 22,1,1,1,5,1,4,5,2,5,4,1,5,1,1,1,44,1,1,1,5,1,4,5,2,5,4,1,5,1,
%T A040490 1,1,44,1,1,1,5,1,4,5,2,5,4,1,5,1,1,1,44,1,1,1,5,1,4,5,2,5,4,1,
%U A040490 5,1,1,1,44,1,1,1,5,1,4,5,2,5,4,1,5,1,1,1,44,1,1,1,5,1,4,5,2,5
%N A040490 Continued fraction for sqrt(513).
%H A040490 Index entries for continued fractions for constants
%p A040490 with(numtheory): Digits:=300: convert(evalf(sqrt(513)),confrac);
%K A040490 nonn,cofr,easy
%O A040490 0,1
%A A040490 njas
%I A040487
%S A040487 22,1,1,2,1,1,44,1,1,2,1,1,44,1,1,2,1,1,44,1,1,2,1,1,44,1,1,2,1,
%T A040487 1,44,1,1,2,1,1,44,1,1,2,1,1,44,1,1,2,1,1,44,1,1,2,1,1,44,1,1,2,
%U A040487 1,1,44,1,1,2,1,1,44,1,1,2,1,1,44,1,1,2,1,1,44,1,1,2,1,1,44,1,1
%N A040487 Continued fraction for sqrt(510).
%H A040487 Index entries for continued fractions for constants
%p A040487 with(numtheory): Digits:=300: convert(evalf(sqrt(510)),confrac);
%K A040487 nonn,cofr,easy
%O A040487 0,1
%A A040487 njas
%I A040486
%S A040486 22,1,1,3,1,1,2,10,1,8,8,1,10,2,1,1,3,1,1,44,1,1,3,1,1,2,10,1,8,
%T A040486 8,1,10,2,1,1,3,1,1,44,1,1,3,1,1,2,10,1,8,8,1,10,2,1,1,3,1,1,44,
%U A040486 1,1,3,1,1,2,10,1,8,8,1,10,2,1,1,3,1,1,44,1,1,3,1,1,2,10,1,8,8
%N A040486 Continued fraction for sqrt(509).
%H A040486 Index entries for continued fractions for constants
%p A040486 with(numtheory): Digits:=300: convert(evalf(sqrt(509)),confrac);
%K A040486 nonn,cofr,easy
%O A040486 0,1
%A A040486 njas
%I A040485
%S A040485 22,1,1,5,1,14,5,1,1,3,4,1,2,1,1,1,10,1,1,1,2,1,4,3,1,1,5,14,1,
%T A040485 5,1,1,44,1,1,5,1,14,5,1,1,3,4,1,2,1,1,1,10,1,1,1,2,1,4,3,1,1,5,
%U A040485 14,1,5,1,1,44,1,1,5,1,14,5,1,1,3,4,1,2,1,1,1,10,1,1,1,2,1,4,3
%N A040485 Continued fraction for sqrt(508).
%H A040485 Index entries for continued fractions for constants
%p A040485 with(numtheory): Digits:=300: convert(evalf(sqrt(508)),confrac);
%K A040485 nonn,cofr,easy
%O A040485 0,1
%A A040485 njas
%I A040484
%S A040484 22,1,1,14,1,1,44,1,1,14,1,1,44,1,1,14,1,1,44,1,1,14,1,1,44,1,1,
%T A040484 14,1,1,44,1,1,14,1,1,44,1,1,14,1,1,44,1,1,14,1,1,44,1,1,14,1,1,
%U A040484 44,1,1,14,1,1,44,1,1,14,1,1,44,1,1,14,1,1,44,1,1,14,1,1,44,1,1
%N A040484 Continued fraction for sqrt(507).
%H A040484 Index entries for continued fractions for constants
%p A040484 with(numtheory): Digits:=300: convert(evalf(sqrt(507)),confrac);
%K A040484 nonn,cofr,easy
%O A040484 0,1
%A A040484 njas
%I A022185
%S A022185 1,1,1,1,22,1,1,463,463,1,1,9724,204646,9724,1,1,204205,
%T A022185 90258610,90258610,204205,1,1,4288306,39804251215,835975245820,
%U A022185 39804251215,4288306,1,1,90054427,17553679074121,7742006555790235
%N A022185 Triangle of Gaussian binomial coefficients [ n,k ] for q = 21.
%D A022185 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%K A022185 nonn,tabl
%O A022185 0,5
%A A022185 njas
%I A015150
%S A015150 1,1,1,1,22,1,1,507,507,1,1,11660,268710,11660,1,1,268181,142135930,
%T A015150 142135930,268181,1,1,6168162,75190175151,1729225724380,75190175151,
%U A015150 6168162,1,1,141867727,39775596486717,21039564578706611
%V A015150 1,1,1,1,-22,1,1,507,507,1,1,-11660,268710,-11660,1,1,268181,142135930,
%W A015150 142135930,268181,1,1,-6168162,75190175151,-1729225724380,75190175151,
%X A015150 -6168162,1,1,141867727,39775596486717,21039564578706611
%N A015150 Triangle of (Gaussian) q-binomial coefficients for q=-23.
%K A015150 sign,done,tabl,easy
%O A015150 0,5
%A A015150 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A040493
%S A040493 22,1,2,1,1,14,1,1,2,1,44,1,2,1,1,14,1,1,2,1,44,1,2,1,1,14,1,1,
%T A040493 2,1,44,1,2,1,1,14,1,1,2,1,44,1,2,1,1,14,1,1,2,1,44,1,2,1,1,14,
%U A040493 1,1,2,1,44,1,2,1,1,14,1,1,2,1,44,1,2,1,1,14,1,1,2,1,44,1,2,1,1
%N A040493 Continued fraction for sqrt(516).
%H A040493 Index entries for continued fractions for constants
%p A040493 with(numtheory): Digits:=300: convert(evalf(sqrt(516)),confrac);
%K A040493 nonn,cofr,easy
%O A040493 0,1
%A A040493 njas
%I A040494
%S A040494 22,1,2,1,4,3,3,2,10,1,14,4,14,1,10,2,3,3,4,1,2,1,44,1,2,1,4,3,
%T A040494 3,2,10,1,14,4,14,1,10,2,3,3,4,1,2,1,44,1,2,1,4,3,3,2,10,1,14,4,
%U A040494 14,1,10,2,3,3,4,1,2,1,44,1,2,1,4,3,3,2,10,1,14,4,14,1,10,2,3,3
%N A040494 Continued fraction for sqrt(517).
%H A040494 Index entries for continued fractions for constants
%p A040494 with(numtheory): Digits:=300: convert(evalf(sqrt(517)),confrac);
%K A040494 nonn,cofr,easy
%O A040494 0,1
%A A040494 njas
%I A040492
%S A040492 22,1,2,3,1,3,1,3,2,1,44,1,2,3,1,3,1,3,2,1,44,1,2,3,1,3,1,3,2,1,
%T A040492 44,1,2,3,1,3,1,3,2,1,44,1,2,3,1,3,1,3,2,1,44,1,2,3,1,3,1,3,2,1,
%U A040492 44,1,2,3,1,3,1,3,2,1,44,1,2,3,1,3,1,3,2,1,44,1,2,3,1,3,1,3,2,1
%N A040492 Continued fraction for sqrt(515).
%H A040492 Index entries for continued fractions for constants
%p A040492 with(numtheory): Digits:=300: convert(evalf(sqrt(515)),confrac);
%K A040492 nonn,cofr,easy
%O A040492 0,1
%A A040492 njas
%I A040491
%S A040491 22,1,2,22,2,1,44,1,2,22,2,1,44,1,2,22,2,1,44,1,2,22,2,1,44,1,2,
%T A040491 22,2,1,44,1,2,22,2,1,44,1,2,22,2,1,44,1,2,22,2,1,44,1,2,22,2,1,
%U A040491 44,1,2,22,2,1,44,1,2,22,2,1,44,1,2,22,2,1,44,1,2,22,2,1,44,1,2
%N A040491 Continued fraction for sqrt(514).
%H A040491 Index entries for continued fractions for constants
%p A040491 with(numtheory): Digits:=300: convert(evalf(sqrt(514)),confrac);
%K A040491 nonn,cofr,easy
%O A040491 0,1
%A A040491 njas
%I A040496
%S A040496 22,1,3,1,1,2,1,2,3,7,3,2,1,2,1,1,3,1,44,1,3,1,1,2,1,2,3,7,3,2,
%T A040496 1,2,1,1,3,1,44,1,3,1,1,2,1,2,3,7,3,2,1,2,1,1,3,1,44,1,3,1,1,2,
%U A040496 1,2,3,7,3,2,1,2,1,1,3,1,44,1,3,1,1,2,1,2,3,7,3,2,1,2,1,1,3,1,44
%N A040496 Continued fraction for sqrt(519).
%H A040496 Index entries for continued fractions for constants
%p A040496 with(numtheory): Digits:=300: convert(evalf(sqrt(519)),confrac);
%K A040496 nonn,cofr,easy
%O A040496 0,1
%A A040496 njas
%I A040495
%S A040495 22,1,3,6,3,1,44,1,3,6,3,1,44,1,3,6,3,1,44,1,3,6,3,1,44,1,3,6,3,
%T A040495 1,44,1,3,6,3,1,44,1,3,6,3,1,44,1,3,6,3,1,44,1,3,6,3,1,44,1,3,6,
%U A040495 3,1,44,1,3,6,3,1,44,1,3,6,3,1,44,1,3,6,3,1,44,1,3,6,3,1,44,1,3
%N A040495 Continued fraction for sqrt(518).
%H A040495 Index entries for continued fractions for constants
%p A040495 with(numtheory): Digits:=300: convert(evalf(sqrt(518)),confrac);
%K A040495 nonn,cofr,easy
%O A040495 0,1
%A A040495 njas
%I A040498
%S A040498 22,1,4,1,2,1,2,8,1,3,3,1,8,2,1,2,1,4,1,44,1,4,1,2,1,2,8,1,3,3,
%T A040498 1,8,2,1,2,1,4,1,44,1,4,1,2,1,2,8,1,3,3,1,8,2,1,2,1,4,1,44,1,4,
%U A040498 1,2,1,2,8,1,3,3,1,8,2,1,2,1,4,1,44,1,4,1,2,1,2,8,1,3,3,1,8,2,1
%N A040498 Continued fraction for sqrt(521).
%H A040498 Index entries for continued fractions for constants
%p A040498 with(numtheory): Digits:=300: convert(evalf(sqrt(521)),confrac);
%K A040498 nonn,cofr,easy
%O A040498 0,1
%A A040498 njas
%I A040497
%S A040497 22,1,4,11,4,1,44,1,4,11,4,1,44,1,4,11,4,1,44,1,4,11,4,1,44,1,4,
%T A040497 11,4,1,44,1,4,11,4,1,44,1,4,11,4,1,44,1,4,11,4,1,44,1,4,11,4,1,
%U A040497 44,1,4,11,4,1,44,1,4,11,4,1,44,1,4,11,4,1,44,1,4,11,4,1,44,1,4
%N A040497 Continued fraction for sqrt(520).
%H A040497 Index entries for continued fractions for constants
%p A040497 with(numtheory): Digits:=300: convert(evalf(sqrt(520)),confrac);
%K A040497 nonn,cofr,easy
%O A040497 0,1
%A A040497 njas
%I A040499
%S A040499 22,1,5,1,1,4,1,1,5,1,44,1,5,1,1,4,1,1,5,1,44,1,5,1,1,4,1,1,5,1,
%T A040499 44,1,5,1,1,4,1,1,5,1,44,1,5,1,1,4,1,1,5,1,44,1,5,1,1,4,1,1,5,1,
%U A040499 44,1,5,1,1,4,1,1,5,1,44,1,5,1,1,4,1,1,5,1,44,1,5,1,1,4,1,1,5,1
%N A040499 Continued fraction for sqrt(522).
%H A040499 Index entries for continued fractions for constants
%p A040499 with(numtheory): Digits:=300: convert(evalf(sqrt(522)),confrac);
%K A040499 nonn,cofr,easy
%O A040499 0,1
%A A040499 njas
%I A040500
%S A040500 22,1,6,1,1,1,4,2,3,14,1,21,1,14,3,2,4,1,1,1,6,1,44,1,6,1,1,1,4,
%T A040500 2,3,14,1,21,1,14,3,2,4,1,1,1,6,1,44,1,6,1,1,1,4,2,3,14,1,21,1,
%U A040500 14,3,2,4,1,1,1,6,1,44,1,6,1,1,1,4,2,3,14,1,21,1,14,3,2,4,1,1,1
%N A040500 Continued fraction for sqrt(523).
%H A040500 Index entries for continued fractions for constants
%p A040500 with(numtheory): Digits:=300: convert(evalf(sqrt(523)),confrac);
%K A040500 nonn,cofr,easy
%O A040500 0,1
%A A040500 njas
%I A040501
%S A040501 22,1,8,5,1,1,1,1,2,1,10,1,2,1,1,1,1,5,8,1,44,1,8,5,1,1,1,1,2,1,
%T A040501 10,1,2,1,1,1,1,5,8,1,44,1,8,5,1,1,1,1,2,1,10,1,2,1,1,1,1,5,8,1,
%U A040501 44,1,8,5,1,1,1,1,2,1,10,1,2,1,1,1,1,5,8,1,44,1,8,5,1,1,1,1,2,1
%N A040501 Continued fraction for sqrt(524).
%H A040501 Index entries for continued fractions for constants
%p A040501 with(numtheory): Digits:=300: convert(evalf(sqrt(524)),confrac);
%K A040501 nonn,cofr,easy
%O A040501 0,1
%A A040501 njas
%I A040502
%S A040502 22,1,10,2,10,1,44,1,10,2,10,1,44,1,10,2,10,1,44,1,10,2,10,1,44,
%T A040502 1,10,2,10,1,44,1,10,2,10,1,44,1,10,2,10,1,44,1,10,2,10,1,44,1,
%U A040502 10,2,10,1,44,1,10,2,10,1,44,1,10,2,10,1,44,1,10,2,10,1,44,1,10
%N A040502 Continued fraction for sqrt(525).
%H A040502 Index entries for continued fractions for constants
%p A040502 with(numtheory): Digits:=300: convert(evalf(sqrt(525)),confrac);
%K A040502 nonn,cofr,easy
%O A040502 0,1
%A A040502 njas
%I A040503
%S A040503 22,1,14,3,4,1,3,2,1,3,1,8,2,1,1,2,2,6,7,2,22,2,7,6,2,2,1,1,2,8,
%T A040503 1,3,1,2,3,1,4,3,14,1,44,1,14,3,4,1,3,2,1,3,1,8,2,1,1,2,2,6,7,2,
%U A040503 22,2,7,6,2,2,1,1,2,8,1,3,1,2,3,1,4,3,14,1,44,1,14,3,4,1,3,2,1
%N A040503 Continued fraction for sqrt(526).
%H A040503 Index entries for continued fractions for constants
%p A040503 with(numtheory): Digits:=300: convert(evalf(sqrt(526)),confrac);
%K A040503 nonn,cofr,easy
%O A040503 0,1
%A A040503 njas
%I A040504
%S A040504 22,1,21,1,44,1,21,1,44,1,21,1,44,1,21,1,44,1,21,1,44,1,21,1,44,
%T A040504 1,21,1,44,1,21,1,44,1,21,1,44,1,21,1,44,1,21,1,44,1,21,1,44,1,
%U A040504 21,1,44,1,21,1,44,1,21,1,44,1,21,1,44,1,21,1,44,1,21,1,44,1,21
%N A040504 Continued fraction for sqrt(527).
%H A040504 Index entries for continued fractions for constants
%p A040504 with(numtheory): Digits:=300: convert(evalf(sqrt(527)),confrac);
%K A040504 nonn,cofr,easy
%O A040504 0,1
%A A040504 njas
%I A040505
%S A040505 22,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,
%T A040505 1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,
%U A040505 44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44,1,44
%N A040505 Continued fraction for sqrt(528).
%H A040505 Index entries for continued fractions for constants
%p A040505 with(numtheory): Digits:=300: convert(evalf(sqrt(528)),confrac);
%K A040505 nonn,cofr,easy
%O A040505 0,1
%A A040505 njas
%I A040478
%S A040478 22,2,1,1,1,1,3,8,1,2,10,1,5,2,14,2,5,1,10,2,1,8,3,1,1,1,1,2,44,
%T A040478 2,1,1,1,1,3,8,1,2,10,1,5,2,14,2,5,1,10,2,1,8,3,1,1,1,1,2,44,2,
%U A040478 1,1,1,1,3,8,1,2,10,1,5,2,14,2,5,1,10,2,1,8,3,1,1,1,1,2,44,2,1
%N A040478 Continued fraction for sqrt(501).
%H A040478 Index entries for continued fractions for constants
%p A040478 with(numtheory): Digits:=300: convert(evalf(sqrt(501)),confrac);
%K A040478 nonn,cofr,easy
%O A040478 0,1
%A A040478 njas
%I A040477
%S A040477 22,2,1,3,2,1,1,10,1,1,2,3,1,2,44,2,1,3,2,1,1,10,1,1,2,3,1,2,44,
%T A040477 2,1,3,2,1,1,10,1,1,2,3,1,2,44,2,1,3,2,1,1,10,1,1,2,3,1,2,44,2,
%U A040477 1,3,2,1,1,10,1,1,2,3,1,2,44,2,1,3,2,1,1,10,1,1,2,3,1,2,44,2,1
%N A040477 Continued fraction for sqrt(500).
%H A040477 Index entries for continued fractions for constants
%p A040477 with(numtheory): Digits:=300: convert(evalf(sqrt(500)),confrac);
%K A040477 nonn,cofr,easy
%O A040477 0,1
%A A040477 njas
%I A040476
%S A040476 22,2,1,21,1,2,44,2,1,21,1,2,44,2,1,21,1,2,44,2,1,21,1,2,44,2,1,
%T A040476 21,1,2,44,2,1,21,1,2,44,2,1,21,1,2,44,2,1,21,1,2,44,2,1,21,1,2,
%U A040476 44,2,1,21,1,2,44,2,1,21,1,2,44,2,1,21,1,2,44,2,1,21,1,2,44,2,1
%N A040476 Continued fraction for sqrt(499).
%H A040476 Index entries for continued fractions for constants
%p A040476 with(numtheory): Digits:=300: convert(evalf(sqrt(499)),confrac);
%K A040476 nonn,cofr,easy
%O A040476 0,1
%A A040476 njas
%I A040480
%S A040480 22,2,2,1,21,1,2,2,44,2,2,1,21,1,2,2,44,2,2,1,21,1,2,2,44,2,2,1,
%T A040480 21,1,2,2,44,2,2,1,21,1,2,2,44,2,2,1,21,1,2,2,44,2,2,1,21,1,2,2,
%U A040480 44,2,2,1,21,1,2,2,44,2,2,1,21,1,2,2,44,2,2,1,21,1,2,2,44,2,2,1
%N A040480 Continued fraction for sqrt(503).
%H A040480 Index entries for continued fractions for constants
%p A040480 with(numtheory): Digits:=300: convert(evalf(sqrt(503)),confrac);
%K A040480 nonn,cofr,easy
%O A040480 0,1
%A A040480 njas
%I A040479
%S A040479 22,2,2,7,14,1,4,22,4,1,14,7,2,2,44,2,2,7,14,1,4,22,4,1,14,7,2,
%T A040479 2,44,2,2,7,14,1,4,22,4,1,14,7,2,2,44,2,2,7,14,1,4,22,4,1,14,7,
%U A040479 2,2,44,2,2,7,14,1,4,22,4,1,14,7,2,2,44,2,2,7,14,1,4,22,4,1,14
%N A040479 Continued fraction for sqrt(502).
%H A040479 Index entries for continued fractions for constants
%p A040479 with(numtheory): Digits:=300: convert(evalf(sqrt(502)),confrac);
%K A040479 nonn,cofr,easy
%O A040479 0,1
%A A040479 njas
%I A040481
%S A040481 22,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,
%T A040481 44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,
%U A040481 44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2,44,2,4,2
%N A040481 Continued fraction for sqrt(504).
%H A040481 Index entries for continued fractions for constants
%p A040481 with(numtheory): Digits:=300: convert(evalf(sqrt(504)),confrac);
%K A040481 nonn,cofr,easy
%O A040481 0,1
%A A040481 njas
%I A058288
%S A058288 22,2,5,1,1,1,1,1,3,2,1,1,3,9,15,25,1,1,5,4,1,2,1,1,50,1,1,1,1,7,1,1,1,
%T A058288 3,6,1,20,10,1,2,10,1,8,2,2,1,1,1,4,1,43,2,2,3,1,2,8,1,1,16,1,4,1,3,1,
%U A058288 1,1,2,1,1,6,1,2,1,1,1,1,1,4,4,1,1,1,9,1,1,105,1,3,6,2,1,1,3,1,3,2,1,1
%N A058288 Continued fraction expansion of Pi^e.
%H A058288 Index entries for continued fractions for constants
%p A058288 cfrac(evalf((evalf(Pi))^(exp(1)),2560),256,'quotients');
%t A058288 ContinuedFraction[ Pi^E, 100]
%o A058288 (PARI) contfrac(Pi^exp(1))
%Y A058288 Cf. A059850.
%K A058288 cofr,nonn,easy
%O A058288 0,1
%A A058288 Robert G. Wilson v (rgwv@kspaint.com), Dec 07 2000
%E A058288 More terms from Jason Earls (jcearls@kskc.net), Jul 12 2001
%I A040482
%S A040482 22,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,
%T A040482 44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,
%U A040482 44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2,44,2,8,2
%N A040482 Continued fraction for sqrt(505).
%H A040482 Index entries for continued fractions for constants
%p A040482 with(numtheory): Digits:=300: convert(evalf(sqrt(505)),confrac);
%K A040482 nonn,cofr,easy
%O A040482 0,1
%A A040482 njas
%I A040483
%S A040483 22,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,
%T A040483 2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,
%U A040483 44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44,2,44
%N A040483 Continued fraction for sqrt(506).
%H A040483 Index entries for continued fractions for constants
%p A040483 with(numtheory): Digits:=300: convert(evalf(sqrt(506)),confrac);
%K A040483 nonn,cofr,easy
%O A040483 0,1
%A A040483 njas
%I A040473
%S A040473 22,3,1,2,4,1,1,2,2,2,1,1,4,2,1,3,44,3,1,2,4,1,1,2,2,2,1,1,4,2,
%T A040473 1,3,44,3,1,2,4,1,1,2,2,2,1,1,4,2,1,3,44,3,1,2,4,1,1,2,2,2,1,1,
%U A040473 4,2,1,3,44,3,1,2,4,1,1,2,2,2,1,1,4,2,1,3,44,3,1,2,4,1,1,2,2,2
%N A040473 Continued fraction for sqrt(496).
%H A040473 Index entries for continued fractions for constants
%p A040473 with(numtheory): Digits:=300: convert(evalf(sqrt(496)),confrac);
%K A040473 nonn,cofr,easy
%O A040473 0,1
%A A040473 njas
%I A040474
%S A040474 22,3,2,2,5,6,5,2,2,3,44,3,2,2,5,6,5,2,2,3,44,3,2,2,5,6,5,2,2,3,
%T A040474 44,3,2,2,5,6,5,2,2,3,44,3,2,2,5,6,5,2,2,3,44,3,2,2,5,6,5,2,2,3,
%U A040474 44,3,2,2,5,6,5,2,2,3,44,3,2,2,5,6,5,2,2,3,44,3,2,2,5,6,5,2,2,3
%N A040474 Continued fraction for sqrt(497).
%H A040474 Index entries for continued fractions for constants
%p A040474 with(numtheory): Digits:=300: convert(evalf(sqrt(497)),confrac);
%K A040474 nonn,cofr,easy
%O A040474 0,1
%A A040474 njas
%I A040475
%S A040475 22,3,6,22,6,3,44,3,6,22,6,3,44,3,6,22,6,3,44,3,6,22,6,3,44,3,6,
%T A040475 22,6,3,44,3,6,22,6,3,44,3,6,22,6,3,44,3,6,22,6,3,44,3,6,22,6,3,
%U A040475 44,3,6,22,6,3,44,3,6,22,6,3,44,3,6,22,6,3,44,3,6,22,6,3,44,3,6
%N A040475 Continued fraction for sqrt(498).
%H A040475 Index entries for continued fractions for constants
%p A040475 with(numtheory): Digits:=300: convert(evalf(sqrt(498)),confrac);
%K A040475 nonn,cofr,easy
%O A040475 0,1
%A A040475 njas
%I A065662
%S A065662 1,22,3,10,94,6,7,20479,175,46,382,12,13,14,15,76,2563,172,5,736,751,
%T A065662 190,1534,24,25,26,27,28,29,30,31,292,2527,655360,1294,343,43,45055,
%U A065662 5632,11,739,748,23,3040,3055,766,6142,48,49,50,51,52,53,54,55,56,57
%N A065662 Permutation of N induced by rotating the node 2 right in the infinite planar binary tree shown at A065658.
%H A065662 Index entries for sequences that are permutations of the natural numbers
%p A065662 [seq(RotateBinFracNodeRight(2,j),j=1..256);]
%Y A065662 The second row of A065658. Inverse of A065663. A065662(n) = A065660(A065666(A065661(n)))
%K A065662 nonn
%O A065662 1,2
%A A065662 Antti.Karttunen@iki.fi Nov 22 2001
%I A040470
%S A040470 22,4,1,10,3,3,10,1,4,44,4,1,10,3,3,10,1,4,44,4,1,10,3,3,10,1,4,
%T A040470 44,4,1,10,3,3,10,1,4,44,4,1,10,3,3,10,1,4,44,4,1,10,3,3,10,1,4,
%U A040470 44,4,1,10,3,3,10,1,4,44,4,1,10,3,3,10,1,4,44,4,1,10,3,3,10,1,4
%N A040470 Continued fraction for sqrt(493).
%H A040470 Index entries for continued fractions for constants
%p A040470 with(numtheory): Digits:=300: convert(evalf(sqrt(493)),confrac);
%K A040470 nonn,cofr,easy
%O A040470 0,1
%A A040470 njas
%I A040471
%S A040471 22,4,2,2,1,2,1,2,2,4,44,4,2,2,1,2,1,2,2,4,44,4,2,2,1,2,1,2,2,4,
%T A040471 44,4,2,2,1,2,1,2,2,4,44,4,2,2,1,2,1,2,2,4,44,4,2,2,1,2,1,2,2,4,
%U A040471 44,4,2,2,1,2,1,2,2,4,44,4,2,2,1,2,1,2,2,4,44,4,2,2,1,2,1,2,2,4
%N A040471 Continued fraction for sqrt(494).
%H A040471 Index entries for continued fractions for constants
%p A040471 with(numtheory): Digits:=300: convert(evalf(sqrt(494)),confrac);
%K A040471 nonn,cofr,easy
%O A040471 0,1
%A A040471 njas
%I A018817
%S A018817 0,0,0,0,0,0,0,0,0,0,22,4,4,4,4,4,4,4,4,44,92,60,20,20,20,20,20,20,136,
%T A018817 268,416,324,216,92,92,92,92,244,412,596,796,676,540,388,220,220,596,
%U A018817 1004,1444,1916,2420,2124,1796,1436,1044,844,1084,1340,1612,1900,2204
%N A018817 Lines through exactly 10 points of an n X n grid of points.
%K A018817 nonn
%O A018817 0,11
%A A018817 dww
%I A040472
%S A040472 22,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,
%T A040472 4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,
%U A040472 44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44,4,44
%N A040472 Continued fraction for sqrt(495).
%H A040472 Index entries for continued fractions for constants
%p A040472 with(numtheory): Digits:=300: convert(evalf(sqrt(495)),confrac);
%K A040472 nonn,cofr,easy
%O A040472 0,1
%A A040472 njas
%I A040469
%S A040469 22,5,1,1,10,1,1,5,44,5,1,1,10,1,1,5,44,5,1,1,10,1,1,5,44,5,1,1,
%T A040469 10,1,1,5,44,5,1,1,10,1,1,5,44,5,1,1,10,1,1,5,44,5,1,1,10,1,1,5,
%U A040469 44,5,1,1,10,1,1,5,44,5,1,1,10,1,1,5,44,5,1,1,10,1,1,5,44,5,1,1
%N A040469 Continued fraction for sqrt(492).
%H A040469 Index entries for continued fractions for constants
%p A040469 with(numtheory): Digits:=300: convert(evalf(sqrt(492)),confrac);
%K A040469 nonn,cofr,easy
%O A040469 0,1
%A A040469 njas
%I A040468
%S A040468 22,6,3,4,8,1,1,1,2,1,1,21,1,1,2,1,1,1,8,4,3,6,44,6,3,4,8,1,1,1,
%T A040468 2,1,1,21,1,1,2,1,1,1,8,4,3,6,44,6,3,4,8,1,1,1,2,1,1,21,1,1,2,1,
%U A040468 1,1,8,4,3,6,44,6,3,4,8,1,1,1,2,1,1,21,1,1,2,1,1,1,8,4,3,6,44,6
%N A040468 Continued fraction for sqrt(491).
%H A040468 Index entries for continued fractions for constants
%p A040468 with(numtheory): Digits:=300: convert(evalf(sqrt(491)),confrac);
%K A040468 nonn,cofr,easy
%O A040468 0,1
%A A040468 njas
%I A040467
%S A040467 22,7,2,1,4,4,4,1,2,7,44,7,2,1,4,4,4,1,2,7,44,7,2,1,4,4,4,1,2,7,
%T A040467 44,7,2,1,4,4,4,1,2,7,44,7,2,1,4,4,4,1,2,7,44,7,2,1,4,4,4,1,2,7,
%U A040467 44,7,2,1,4,4,4,1,2,7,44,7,2,1,4,4,4,1,2,7,44,7,2,1,4,4,4,1,2,7
%N A040467 Continued fraction for sqrt(490).
%H A040467 Index entries for continued fractions for constants
%p A040467 with(numtheory): Digits:=300: convert(evalf(sqrt(490)),confrac);
%K A040467 nonn,cofr,easy
%O A040467 0,1
%A A040467 njas
%I A040466
%S A040466 22,8,1,4,1,1,1,3,2,1,2,14,2,1,2,3,1,1,1,4,1,8,44,8,1,4,1,1,1,3,
%T A040466 2,1,2,14,2,1,2,3,1,1,1,4,1,8,44,8,1,4,1,1,1,3,2,1,2,14,2,1,2,3,
%U A040466 1,1,1,4,1,8,44,8,1,4,1,1,1,3,2,1,2,14,2,1,2,3,1,1,1,4,1,8,44,8
%N A040466 Continued fraction for sqrt(489).
%H A040466 Index entries for continued fractions for constants
%p A040466 with(numtheory): Digits:=300: convert(evalf(sqrt(489)),confrac);
%K A040466 nonn,cofr,easy
%O A040466 0,1
%A A040466 njas
%I A058402
%S A058402 1,22,8,588,376,56,19656,17024,4576,384,801360,848096,313504,48256,
%T A058402 2624,38797920,47494272,21685888,4643072,468608,17920,2181332160,
%U A058402 2986217856,1590913920,424509952,60136448,4307456,122368,139864717440
%N A058402 Coefficient triangle of polynomials (rising powers) related to Pell number convolutions. Companion triangle is A058403.
%C A058402 The row polynomials are p(k,x):=sum(a(k,m)*x^m,m=0..k), k=0,1,2,...
%C A058402 The k-th convolution of P0(n):= A000129(n+1) n >= 0, (Pell numbers starting with P0(0)=1) with itself is Pk(n):= A054456(n+k,k) = (p(k-1,n)*(n+1)*2*P0(n+1) + q(k-1,n)*(n+2)*P0(n))/(k!*8^k)), k=1,2,..., where the companion polynomials q(k,n):= sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A058403(k,m).
%F A058402 Recursion for row polynomials defined in the comments: p(k,n)= 4*(n+2)*p(k-1,n+1)+2*(n+2*(k+1))*p(k-1,n)+(n+3)*q(k-1,n); q(k,n)= 4*(n+1)*p(k-1,n+1)+2*(n+2*(k+1))*q(k-1,n), k >= 1.
%e A058402 k=2: P2(n)=((22+8*n)*(n+1)*2*P0(n+1)+(20+8*n)*(n+2)*P0(n))/128, cf. A054457.
%e A058402 1; 22,8; 588,376,56; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).
%Y A058402 Cf. A000129, A054456, A058403.
%K A058402 nonn,tabl
%O A058402 0,2
%A A058402 Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de), Dec 11 2000
%I A033342
%S A033342 22,11,7,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,
%T A033342 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A033342 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A033342 [ 22/n ].
%K A033342 easy,nonn
%O A033342 1,1
%A A033342 Jeff Burch (jmburch@osprey.smcm.edu)
%I A040465
%S A040465 22,11,44,11,44,11,44,11,44,11,44,11,44,11,44,11,44,11,44,11,44,
%T A040465 11,44,11,44,11,44,11,44,11,44,11,44,11,44,11,44,11,44,11,44,11,
%U A040465 44,11,44,11,44,11,44,11,44,11,44,11,44,11,44,11,44,11,44,11,44
%N A040465 Continued fraction for sqrt(488).
%H A040465 Index entries for continued fractions for constants
%p A040465 with(numtheory): Digits:=300: convert(evalf(sqrt(488)),confrac);
%K A040465 nonn,cofr,easy
%O A040465 0,1
%A A040465 njas
%I A033967
%S A033967 1,22,11,232,116,58,29,610,305,6406,3203,67264,33632,16816,
%T A033967 8408,4204,2102,1051,22072,11036,5518,2759,57940,28970,
%U A033967 14485,304186,152093,3193954,1596977,33536518,16768259
%N A033967 Trajectory of 1 under map n->21n+1 if n odd, n->n/2 if n even
%K A033967 nonn
%O A033967 0,2
%A A033967 njas
%I A040464
%S A040464 22,14,1,2,4,1,1,3,2,5,1,6,1,1,21,1,1,6,1,5,2,3,1,1,4,2,1,14,44,
%T A040464 14,1,2,4,1,1,3,2,5,1,6,1,1,21,1,1,6,1,5,2,3,1,1,4,2,1,14,44,14,
%U A040464 1,2,4,1,1,3,2,5,1,6,1,1,21,1,1,6,1,5,2,3,1,1,4,2,1,14,44,14,1
%N A040464 Continued fraction for sqrt(487).
%H A040464 Index entries for continued fractions for constants
%p A040464 with(numtheory): Digits:=300: convert(evalf(sqrt(487)),confrac);
%K A040464 nonn,cofr,easy
%O A040464 0,1
%A A040464 njas
%I A022978
%S A022978 22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,
%T A022978 1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,
%U A022978 21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38
%V A022978 22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,
%W A022978 1,0,-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,-11,-12,-13,-14,-15,-16,-17,-18,-19,-20,
%X A022978 -21,-22,-23,-24,-25,-26,-27,-28,-29,-30,-31,-32,-33,-34,-35,-36,-37,-38
%N A022978 22-n.
%K A022978 sign,done
%O A022978 0,1
%A A022978 njas
%I A023464
%S A023464 22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,
%T A023464 1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,
%U A023464 21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38
%V A023464 -22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,
%W A023464 -1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,
%X A023464 21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38
%N A023464 n-22.
%K A023464 sign,done
%O A023464 0,1
%A A023464 njas
%I A010861
%S A010861 22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,
%T A010861 22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,
%U A010861 22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22
%N A010861 Constant sequence.
%C A010861 Describe the previous term! (method A - initial term is 22).
%Y A010861 Cf. A005150, A005151.
%K A010861 nonn
%O A010861 0,1
%A A010861 njas
%I A040463
%S A040463 22,22,44,22,44,22,44,22,44,22,44,22,44,22,44,22,44,22,44,22,44,
%T A040463 22,44,22,44,22,44,22,44,22,44,22,44,22,44,22,44,22,44,22,44,22,
%U A040463 44,22,44,22,44,22,44,22,44,22,44,22,44,22,44,22,44,22,44,22,44
%N A040463 Continued fraction for sqrt(486).
%H A040463 Index entries for continued fractions for constants
%p A040463 with(numtheory): Digits:=300: convert(evalf(sqrt(486)),confrac);
%K A040463 nonn,cofr,easy
%O A040463 0,1
%A A040463 njas
%I A022356
%S A022356 0,22,22,44,66,110,176,286,462,748,1210,1958,3168,5126,
%T A022356 8294,13420,21714,35134,56848,91982,148830,240812,389642,
%U A022356 630454,1020096,1650550,2670646,4321196,6991842,11313038
%N A022356 Fibonacci sequence beginning 0 22.
%K A022356 nonn
%O A022356 0,2
%A A022356 njas
%I A048429
%S A048429 22,22,666,666,1186811,1633361,7327237,204656402,971292179,32418381423,
%T A048429 185295592581,5760554550675,6259909099526,148967656769841
%N A048429 Smallest numerator of fraction using palindromes that approximates 'Pi' with at least n decimals.
%e A048429 E.g. n=4 -> 666/212 = 3.1415... correct to four decimals.
%Y A048429 Denominators in A048430.
%K A048429 nonn,frac,base
%O A048429 1,1
%A A048429 Patrick De Geest (pdg@worldofnumbers.com), Apr 1999.
%I A004463
%S A004463 22,23,20,21,18,19,16,17,30,31,28,29,26,27,24,25,6,7,4,
%T A004463 5,2,3,0,1,14,15,12,13,10,11,8,9,54,55,52,53,50,51,48,49,
%U A004463 62,63,60,61,58,59,56,57,38,39,36,37,34,35,32,33,46,47
%N A004463 Nimsum n + 22.
%D A004463 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
%D A004463 J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
%H A004463 Index entries for sequences related to Nim-sums
%K A004463 nonn
%O A004463 0,1
%A A004463 njas
%I A004511
%S A004511 22,23,21,25,26,24,19,20,18,4,5,3,7,8,6,1,2,0,13,14,12,
%T A004511 16,17,15,10,11,9,49,50,48,52,53,51,46,47,45,31,32,30,34,
%U A004511 35,33,28,29,27,40,41,39,43,44,42,37,38,36,76,77,75,79
%N A004511 Tersum n + 22.
%F A004511 Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g. 5 + 8 = "21" + "22" = "10" = 1.
%K A004511 nonn
%O A004511 0,1
%A A004511 njas
%I A058814
%S A058814 1,22,23,24,266,267,268,2712,2713,27175,27176,271819,271820,271821,
%T A058814 2718272,2718273
%N A058814 n divides the number of digits of n!.
%t A058814 Do[ If[ Mod[ Length[ IntegerDigits[ n! ] ], n ] == 0, Print[ n ] ], {n, 1, 10^5} ]
%K A058814 nonn,base
%O A058814 1,2
%A A058814 Robert G. Wilson v (rgwv@kspaint.com), Jan 03 2001
%I A034895
%S A034895 22,23,25,27,32,33,35,37,52,53,55,57,72,73,75,77,111,113,117,119,131,
%T A034895 137,171,173,179,197,311,317,371,411,413,417,431,437,471,473,611,617,
%U A034895 671,711,713,719,731,1013,1031,1037,1073,1079,1097,1379,1397,1499,1673
%N A034895 Dropping any digit gives a prime number.
%K A034895 nonn,base,nice
%O A034895 0,1
%A A034895 Erich Friedman (erich.friedman@stetson.edu)
%I A022392
%S A022392 1,22,23,45,68,113,181,294,475,769,1244,2013,3257,5270,
%T A022392 8527,13797,22324,36121,58445,94566,153011,247577,400588,
%U A022392 648165,1048753,1696918,2745671,4442589,7188260,11630849
%N A022392 Fibonacci sequence beginning 1 22.
%K A022392 nonn
%O A022392 0,2
%A A022392 njas
%I A041976
%S A041976 22,23,45,68,113,746,859,12772,51947,64719,1411046,1475765,
%T A041976 7314106,103873249,111187355,770997379,882184734,1653182113,
%U A041976 2535366847,4188548960,186831521087,191020070047,377851591134
%N A041976 Numerators of continued fraction convergents to sqrt(511).
%Y A041976 Cf. A041977.
%K A041976 nonn,cofr,easy
%O A041976 0,1
%A A041976 njas
%I A041978
%S A041978 22,23,45,68,181,1154,12875,78404,169683,248087,417770,
%T A041978 665857,29715478,30381335,60096813,90478148,241053109,1536796802,
%U A041978 17145817931,104411704388,225969226707,330380931095,556350157802
%N A041978 Numerators of continued fraction convergents to sqrt(512).
%Y A041978 Cf. A041979.
%K A041978 nonn,cofr,easy
%O A041978 0,1
%A A041978 njas
%I A041980
%S A041980 22,23,45,68,385,453,2197,11438,25073,136803,572285,709088,
%T A041980 4117725,4826813,8944538,13771351,614883982,628655333,1243539315,
%U A041980 1872194648,10604512555,12476707203,60511341367,315033414038
%N A041980 Numerators of continued fraction convergents to sqrt(513).
%Y A041980 Cf. A041981.
%K A041980 nonn,cofr,easy
%O A041980 0,1
%A A041980 njas
%I A041974
%S A041974 22,23,45,113,158,271,12082,12353,24435,61223,85658,146881,
%T A041974 6548422,6695303,13243725,33182753,46426478,79609231,3549232642,
%U A041974 3628841873,7178074515,17984990903,25163065418,43148056321
%N A041974 Numerators of continued fraction convergents to sqrt(510).
%Y A041974 Cf. A041975.
%K A041974 nonn,cofr,easy
%O A041974 0,1
%A A041974 njas
%I A041972
%S A041972 22,23,45,158,203,361,925,9611,10536,93899,761728,855627,
%T A041972 9317998,19491623,28809621,48301244,173713353,222014597,
%U A041972 395727950,17634044397,18029772347,35663816744,125021222579
%N A041972 Numerators of continued fraction convergents to sqrt(509).
%Y A041972 Cf. A041973.
%K A041972 nonn,cofr,easy
%O A041972 0,1
%A A041972 njas
%I A041970
%S A041970 22,23,45,248,293,4350,22043,26393,48436,171701,735240,
%T A041970 906941,2549122,3456063,6005185,9461248,100617665,110078913,
%U A041970 210696578,320775491,852247560,1173023051,5544339764,17806042343
%N A041970 Numerators of continued fraction convergents to sqrt(508).
%Y A041970 Cf. A041971.
%K A041970 nonn,cofr,easy
%O A041970 0,1
%A A041970 njas
%I A041968
%S A041968 22,23,45,653,698,1351,60142,61493,121635,1764383,1886018,
%T A041968 3650401,162503662,166154063,328657725,4767362213,5096019938,
%U A041968 9863382151,439084834582,448948216733,888033051315,12881410935143
%N A041968 Numerators of continued fraction convergents to sqrt(507).
%Y A041968 Cf. A041969.
%K A041968 nonn,cofr,easy
%O A041968 0,1
%A A041968 njas
%I A041986
%S A041986 22,23,68,91,159,2317,2476,4793,12062,16855,753682,770537,
%T A041986 2294756,3065293,5360049,78105979,83466028,161572007,406610042,
%U A041986 568182049,25406620198,25974802247,77356224692,103331026939
%N A041986 Numerators of continued fraction convergents to sqrt(516).
%Y A041986 Cf. A041987.
%K A041986 nonn,cofr,easy
%O A041986 0,1
%A A041986 njas
%I A041988
%S A041988 22,23,68,91,432,1387,4593,10573,110323,120896,1802867,
%T A041988 7332364,104455963,111788327,1222339233,2556466793,8891739612,
%U A041988 29231685629,125818482128,155050167757,435918817642,590968985399
%N A041988 Numerators of continued fraction convergents to sqrt(517).
%Y A041988 Cf. A041989.
%K A041988 nonn,cofr,easy
%O A041988 0,1
%A A041988 njas
%I A041984
%S A041984 22,23,68,227,295,1112,1407,5333,12073,17406,777937,795343,
%T A041984 2368623,7901212,10269835,38710717,48980552,185652373,420285298,
%U A041984 605937671,27081542822,27687480493,82456503808,275056991917
%N A041984 Numerators of continued fraction convergents to sqrt(515).
%Y A041984 Cf. A041985.
%K A041984 nonn,cofr,easy
%O A041984 0,1
%A A041984 njas
%I A041982
%S A041982 22,23,68,1519,3106,4625,206606,211231,629068,14050727,
%T A041982 28730522,42781249,1911105478,1953886727,5818878932,129969223231,
%U A041982 265757325394,395726548625,17677725464894,18073452013519
%N A041982 Numerators of continued fraction convergents to sqrt(514).
%Y A041982 Cf. A041983.
%K A041982 nonn,cofr,easy
%O A041982 0,1
%A A041982 njas
%I A041992
%S A041992 22,23,91,114,205,524,729,1982,6675,48707,152796,354299,
%T A041992 507095,1368489,1875584,3244073,11607803,14851876,665090347,
%U A041992 679942223,2704917016,3384859239,6089776255,15564411749
%N A041992 Numerators of continued fraction convergents to sqrt(519).
%Y A041992 Cf. A041993.
%K A041992 nonn,cofr,easy
%O A041992 0,1
%A A041992 njas
%I A041990
%S A041990 22,23,91,569,1798,2367,105946,108313,430885,2693623,8511754,
%T A041990 11205377,501548342,512753719,2039809499,12751610713,40294641638,
%U A041990 53046252351,2374329745082,2427375997433,9656457737381
%N A041990 Numerators of continued fraction convergents to sqrt(518).
%Y A041990 Cf. A041991.
%K A041990 nonn,cofr,easy
%O A041990 0,1
%A A041990 njas
%I A041996
%S A041996 22,23,114,137,388,525,1438,12029,13467,52430,170757,223187,
%T A041996 1956253,4135693,6091946,16319585,22411531,105965709,128377240,
%U A041996 5754564269,5882941509,29286330305,35169271814,99624873933
%N A041996 Numerators of continued fraction convergents to sqrt(521).
%Y A041996 Cf. A041997.
%K A041996 nonn,cofr,easy
%O A041996 0,1
%A A041996 njas
%I A041994
%S A041994 22,23,114,1277,5222,6499,291178,297677,1481886,16598423,
%T A041994 67875578,84474001,3784731622,3869205623,19261554114,215746300877,
%U A041994 882246757622,1097993058499,49193941331578,50291934390077
%N A041994 Numerators of continued fraction convergents to sqrt(520).
%Y A041994 Cf. A041995.
%K A041994 nonn,cofr,easy
%O A041994 0,1
%A A041994 njas
%I A041998
%S A041998 22,23,137,160,297,1348,1645,2993,16610,19603,879142,898745,
%T A041998 5372867,6271612,11644479,52849528,64494007,117343535,651211682,
%U A041998 768555217,34467641230,35236196447,210648623465,245884819912
%N A041998 Numerators of continued fraction convergents to sqrt(522).
%Y A041998 Cf. A041999.
%K A041998 nonn,cofr,easy
%O A041998 0,1
%A A041998 njas
%I A042000
%S A042000 22,23,160,183,343,526,2447,5420,18707,267318,286025,6273843,
%T A042000 6559868,98111995,300895853,699903701,3100510657,3800414358,
%U A042000 6900925015,10701339373,71108961253,81810300626,3670762188797
%N A042000 Numerators of continued fraction convergents to sqrt(523).
%Y A042000 Cf. A042001.
%K A042000 nonn,cofr,easy
%O A042000 0,1
%A A042000 njas
%I A042002
%S A042002 22,23,206,1053,1259,2312,3571,5883,15337,21220,227537,
%T A042002 248757,725051,973808,1698859,2672667,4371526,24530297,
%U A042002 200613902,225144199,10106958658,10332102857,92763781514
%N A042002 Numerators of continued fraction convergents to sqrt(524).
%Y A042002 Cf. A042003.
%K A042002 nonn,cofr,easy
%O A042002 0,1
%A A042002 njas
%I A042004
%S A042004 22,23,252,527,5522,6049,271678,277727,3048948,6375623,
%T A042004 66805178,73180801,3286760422,3359941223,36886172652,77132286527,
%U A042004 808209037922,885341324449,39763227313678,40648568638127
%N A042004 Numerators of continued fraction convergents to sqrt(525).
%Y A042004 Cf. A042005.
%K A042004 nonn,cofr,easy
%O A042004 0,1
%A A042004 njas
%I A042006
%S A042006 22,23,344,1055,4564,5619,21421,48461,69882,258107,327989,
%T A042006 2882019,6092027,8974046,15066073,39106192,93278457,598776934,
%U A042006 4284716995,9168210924,205985357323,421138925570,3153957836313
%N A042006 Numerators of continued fraction convergents to sqrt(526).
%Y A042006 Cf. A042007.
%K A042006 nonn,cofr,easy
%O A042006 0,1
%A A042006 njas
%I A042008
%S A042008 22,23,505,528,23737,24265,533302,557567,25066250,25623817,
%T A042008 563166407,588790224,26469936263,27058726487,594703192490,
%U A042008 621761918977,27952227627478,28573989546455,628006008103033
%N A042008 Numerators of continued fraction convergents to sqrt(527).
%Y A042008 Cf. A042009.
%K A042008 nonn,cofr,easy
%O A042008 0,1
%A A042008 njas
%I A042009
%S A042009 1,1,22,23,1034,1057,23231,24288,1091903,1116191,24531914,
%T A042009 25648105,1153048534,1178696639,25905677953,27084374592,
%U A042009 1217618160001,1244702534593,27356371386454,28601073921047
%N A042009 Denominators of continued fraction convergents to sqrt(527).
%Y A042009 Cf. A042008.
%K A042009 nonn,cofr,easy
%O A042009 0,3
%A A042009 njas
%I A042010
%S A042010 22,23,1034,1057,47542,48599,2185898,2234497,100503766,
%T A042010 102738263,4620987338,4723725601,212464913782,217188639383,
%U A042010 9768765046634,9985953686017,449150727231382,459136680917399
%N A042010 Numerators of continued fraction convergents to sqrt(528).
%Y A042010 Cf. A042011.
%K A042010 nonn,cofr,easy
%O A042010 0,1
%A A042010 njas
%I A067189
%S A067189 22,24,26,30,40,44,52,56,62,98,128
%N A067189 Numbers which can be expressed as the sum of two primes in exactly three ways.
%H A067189 Index entries for sequences related to Goldbach conjecture
%e A067189 26 is a term as 26 = 23+3 =19+7 =13+13 are all the three ways to express 26 as a sum of two primes.
%Y A067189 Cf. A023036, A067187-A067191, A066722.
%K A067189 nonn,fini,full
%O A067189 1,1
%A A067189 Amarnath Murthy (amarnath_murthy@yahoo.com), Jan 10 2002
%E A067189 Extended by Peter Bertok (peter@bertok.com), who finds that there are no other terms below 10000 and conjectures there are no further terms in this sequence and A067188, A067190, etc. - Jan 13, 2002
%E A067189 Richard Guy (Jan 14, 2002) remarks: "I believe that these conjectures follow from a more general one by Hardy & Littlewood (probably in Some problems of `partitio numerorum' III, on the expression of a number as a sum of primes, Acta Math 44(1922) 1-70)".
%I A030593
%S A030593 22,24,26,71,74,77,125,128,131,179,182,185,196,199,202,205,208,
%T A030593 211,214,217,220,223,226,229,232,233,235,236,238,239,241,244,
%U A030593 247,287,290,293,354,358,362,426,430,434,498,502,506,570,574
%N A030593 Position of n-th 4 in A030588.
%K A030593 nonn
%O A030593 1,1
%A A030593 Clark Kimberling, ck6@cedar.evansville.edu
%I A061411
%S A061411 1,22,24,65,110,112,135,137,166,167,218,219,220,228,229,239,257,280,
%T A061411 310,344,345,346,398,399,403,404,405,430,439,440,475,494,504,505,508,
%U A061411 522,524,534,535,536,557,559,592,619,620,624,625,626,679,703,705,706
%N A061411 m.2^n-1 is not prime for all coefficients m in the range 0<=m<=n.
%Y A061411 Cf. A061410-A061415.
%K A061411 nonn
%O A061411 0,2
%A A061411 Patrick De Geest (pdg@worldofnumbers.com), May 2001.
%I A053779
%S A053779 1,22,24,167,202,226,1443,2380,3190,3952,4220,16827,26304,37612,40813,
%T A053779 213501,376524,1920079,2061085,2635057,3463613,4268588,16513206,
%U A053779 68101132,166428703,207224360,403784450,421279478,1384813481
%N A053779 Sum of first n composite numbers is palindromic.
%C A053779 Sequence is 4 + 6 + 8 + 9 + 10 + 12 + ... + z. For values of z see A057959.
%H A053779 P. De Geest, Palindromic Sums
%H A053779 C. Rivera, PrimePuzzle 89
%Y A053779 Cf. A002808, A002113, A053780.
%K A053779 base,nonn
%O A053779 1,2
%A A053779 G. L. Honaker, Jr. (curios@bvub.com), Mar 29 2000
%E A053779 More terms from G. L. Honaker, Jr. (curios@bvub.com), Mar 29 2000. Further terms added by Patrick De Geest (pdg@worldofnumbers.com), 11/2000.
%I A034304
%S A034304 22,25,27,32,33,35,52,55,57,72,75,77,111,117,119,171,371,411,413,417,
%T A034304 437,471,473,611,671,711,713,731,1379,1397,1673,1739,1937,1991,2233,
%U A034304 2277,2571,2577,2811,3113,3131,3173,3311,3317,3479,4199,4331,4433,4439
%N A034304 Non-prime; prime if any digit deleted.
%Y A034304 Cf. A034302-A034305.
%K A034304 base,nonn,nice
%O A034304 1,1
%A A034304 dww (ctron.com)
%I A066737
%S A066737 22,25,27,32,33,35,52,55,57,72,75,77,112,115,117,132,133,135,172,175,
%T A066737 177,192,195
%N A066737 Composite numbers that are concatenations of primes.
%e A066737 72 is the concatenation of primes 7 and 2. 132 is the concatenation of primes 13 and 2.
%K A066737 base,easy,nonn
%O A066737 1,1
%A A066737 Joseph L. Pe (joseph_l_pe@hotmail.com), Jan 15 2002
%I A061371
%S A061371 22,25,27,32,33,35,52,55,57,72,75,77,222,225,232,235,237,252,253,255,
%T A061371 272,273,275,322,323,325,327,332,333,335,352,355,357,372,375,377,522,
%U A061371 525,527,532,533,535,537,552,553,555,572,573,575,722,723,725,732,735
%N A061371 Composite numbers with all prime digits.
%e A061371 a(5) = 35 is composite with digits 3 and 5 which are primes.
%Y A061371 Cf. A061372.
%K A061371 nonn,base
%O A061371 0,1
%A A061371 Amarnath Murthy (amarnath_murthy@yahoo.com), May 02 2001
%E A061371 Corrected and extended by Larry Reeves (larryr@acm.org), May 08 2001
%I A045096
%S A045096 22,25,37,70,73,82,88,91,94,97,100,103,109,118,121,133,145,148,151,157,
%T A045096 181,214,217,229,262,265,274,280,283,286,289,292,295,301,310,313,322,
%U A045096 328,331,334,352,355,364,367,370,376,379,382
%N A045096 In base 4 representation the numbers of 1's and 2's are 2 and 1, respectively.
%K A045096 nonn,base
%O A045096 1,1
%A A045096 Clark Kimberling, ck6@cedar.evansville.edu
%I A066059
%S A066059 22,26,28,35,37,41,46,47,49,60,61,67,75,77,78,84,86,89,90,94,95,97,105,
%T A066059 106,108,110,116,120,122,124,125,131,135,139,141,147,149,152,155,157,
%U A066059 158,163,164,166,169,172,174,177,180,182,185,186,190,191,193,197,199
%N A066059 Integers such that the 'Reverse and Add!' algorithm in base 2 (cf. A062128) does not lead to a palindrome.
%C A066059 The analogue of A023108 in base 2. Program: (ARIBAS): For function b2reverse see A066057; function a066059(mx,stop: integer); var k,c,m,rev: integer; begin for k := 1 to mx do c:= 0; m := k; rev := b2reverse(m); while m <> rev and c < stop do inc(c); m := m + rev; rev := b2reverse(m); end; if c >= stop then write(k," "); end; end; end; a066059(210,300).
%H A066059 K. Brockhaus, On the'Reverse and Add!' algorithm in base 2
%Y A066059 Cf. A062128, A023108, A062130, A033865, A058042, A061561, A066057.
%K A066059 base,nonn
%O A066059 0,1
%A A066059 Klaus Brockhaus (klaus-brockhaus@t-online.de), Dec 04 2001
%I A063940
%S A063940 22,26,33,34,38,39,44,46,51,52,55,57,58,62,65,66,68,69
%N A063940 Composite n such that Ramanujan's function tau(n) (A000594) is not divisible by n.
%e A063940 Tau(22) = 18643272 which Mod( Tau(22), 22) == 10.
%t A063940 Select[ Range[ 70 ], Mod[ CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 70} ] ], 70 ], x ][ [ # ] ], # ] != 0 && ! PrimeQ[ # ] & ]
%Y A063940 Cf. A063938.
%K A063940 nonn
%O A063940 1,1
%A A063940 Robert G. Wilson v (rgwv@kspaint.com), Aug 31 2001
%I A046442
%S A046442 22,26,33,34,38,39,46,51,55,57,58,62,65,69,74,77,82,85,86,87,91,93,94,
%T A046442 95,106,111,115,118,119,122,123,129,133,134,141,142,145,146,155,158,
%U A046442 159,161,166,177,178,183,185,194,201,202,203,205,206,213,214,215,217
%N A046442 Composite numbers whose 2 prime factors are distinct in length.
%e A046442 E.g. 203 because 203 = 7 * 29 = (1)(2).
%K A046442 nonn,base
%O A046442 0,1
%A A046442 Patrick De Geest (pdg@worldofnumbers.com), Jul 1998.
%I A038355
%S A038355 22,27,58,63,94,99,112,118,124,132,133,134,137,142,147,153,159,162,163,
%T A038355 164,167,177,202,207,238,243,274,279,310,315,328,334,340,348,349,350,
%U A038355 353,358,363,369,375,378,379,380,383,393,418,423,454,459,490,495
%N A038355 Representation in base 6 has same nonzero number of 3's and 4's.
%K A038355 nonn,base,easy
%O A038355 0,1
%A A038355 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043128
%S A043128 22,27,58,63,94,99,130,132,162,171,202,207,238,243,274,279,310,
%T A043128 315,346,348,378,387,418,423,454,459,490,495,526,531,562,564,
%U A043128 594,603,634,639,670,675,706,711,742,747,778,780,792,850,855
%N A043128 3 and 4 occur juxtaposed in the base 6 representation of n but not of n-1.
%K A043128 nonn,base
%O A043128 1,1
%A A043128 Clark Kimberling, ck6@cedar.evansville.edu
%I A043908
%S A043908 22,27,58,63,94,99,130,137,167,171,202,207,238,243,274,279,310,
%T A043908 315,346,353,383,387,418,423,454,459,490,495,526,531,562,569,
%U A043908 599,603,634,639,670,675,706,711,742,747,778,785,827,850,855
%N A043908 3 and 4 occur juxtaposed in the base 6 representation of n but not of n+1.
%K A043908 nonn,base
%O A043908 1,1
%A A043908 Clark Kimberling, ck6@cedar.evansville.edu
%I A067035
%S A067035 0,22,33,44,55,66,77,88,99,1111,2552,2662,2772,2882,2992,3663,3773,
%T A067035 3883,3993,4774,4884,4994,5885,5995,6886,6996,7887,7997,8888,8998,9889,
%U A067035 9999
%N A067035 n sets a new record for the number of k's such that n = k + reverse(k).
%C A067035 A067036 gives the corresponding records. It is very likely that 999999 sets another record, since there are 900 integers k such that 999999 = k + reverse(k), but probably there are a few intervening records.
%e A067035 33 belongs to the sequence because for three integers k (cf. A067032) we have 33 = k + reverse(k), and for m < 33 there are at most two integers j such that m = j + reverse(j).
%Y A067035 Cf. A067030, A067031, A067032, A067033, A067034, A067036.
%K A067035 base,more,nonn
%O A067035 0,2
%A A067035 Klaus Brockhaus (klaus-brockhaus@t-online.de), Dec 29 2001
%I A026044
%S A026044 22,33,49,70,97,132,176,229,292,367,455,556,671,802,950,1115,1298,1501,1725,1970,
%T A026044 2237,2528,2844,3185,3552,3947,4371,4824,5307,5822,6370,6951,7566,8217,8905,9630,
%U A026044 10393,11196,12040,12925,13852,14823,15839,16900,18007,19162,20366,21619,22922,24277
%N A026044 a(n) = (d(n)-r(n))/2, where d = A026043 and r is the periodic sequence with fund. period (1,1,0,0).
%K A026044 nonn
%O A026044 5,1
%A A026044 Clark Kimberling (ck6@cedar.evansville.edu)
%I A067087
%S A067087 22,33,55,77,1111,1331,1771,1991,2332,2992,3113,3773,4114,4334,4774,
%T A067087 5335,5995,6116,6776,7117,7337,7997,8338,8998,9779,101101,103301,
%U A067087 107701,109901,113311,127721,131131,137731,139931,149941,151151,157751
%N A067087 Concatenation of n-th prime and its reverse.
%F A067087 Every term of the sequence is divisible by 11.
%t A067087 Table[ ToExpression[ StringJoin[ ToString[Prime[n]], StringReverse[ ToString[ Prime[n]]]]], {n, 1, 40} ]
%K A067087 easy,nonn,base
%O A067087 1,1
%A A067087 Amarnath Murthy (amarnath_murthy@yahoo.com), Jan 07 2002
%E A067087 More terms from Robert G. Wilson v (rgwv@kspaint.com), Jan 09 2002
%I A020151
%S A020151 22,33,91,154,165,169,265,341,385,451,481,553,561,638,946,1027,1045,1065,
%T A020151 1105,1183,1271,1729,1738,1749,2059,2321,2465,2501,2701,2821,2926,3097,
%U A020151 3445,4033,4081,4345,4371,4681,5005,5149,6253,6369,6533,6541,7189,7267
%N A020151 Pseudoprimes to base 23.
%H A020151 Index entries for sequences related to pseudoprimes
%K A020151 nonn
%O A020151 1,1
%A A020151 dww
%I A061561
%S A061561 22,35,84,105,180,225,360,405,744,837,1488,1581,3024,3213,6048,6237,
%T A061561 12192,12573,24384,24765,48960,49725,97920,98685,196224,197757,392448,
%U A061561 393981,785664,788733,1571328,1574397,3144192,3150333,6288384,6294525
%N A061561 Sequence A058042 written in base 10.
%H A061561 K. Brockhaus, On the 'Reverse and Add!' algorithm in base 2
%o A061561 (ARIBAS) var m,c,rev: integer; end; m := 22; c := 1; write(m," "); rev := bit_reverse(m); while m <> rev and c < 40 do inc(c); m := m + rev; write(m," "); rev := bit_reverse(m); end;
%Y A061561 Cf. A058042.
%K A061561 nonn,base
%O A061561 1,1
%A A061561 njas, May 18 2001
%E A061561 More terms from Klaus Brockhaus (klaus-brockhaus@t-online.de), May 27 2001
%I A063252
%S A063252 22,38,46,54,70,78,86,92,94,102,110,134,142,150,156,158,166,174,182,
%T A063252 188,189,190,198,206,214,220,222,230,238,262,270,278,284,286,294,302,
%U A063252 310,316,317,318,326,334,342,348,350,358,366,374,376,378,380,381,382
%N A063252 Numbers for which a mixed L/R binary rotation sequence will reach a fixed point sooner than either a pure only-left or only-right iteration.
%C A063252 See A063251 for more.
%e A063252 a(1)=22 because n=22 is the first n where A063251(n) that is less than both A048881(n) and A063250(n). Similarly, 189 appears because A048881(189)=5 and A063250(189)=7, but A063251(189) is just 4.
%Y A063252 A063251.
%K A063252 base,easy,nonn
%O A063252 1,1
%A A063252 Marc LeBrun (mlb@well.com), Jul 11 2001
%I A039373
%S A039373 22,38,103,119,166,175,193,198,199,201,203,204,205,206,211,220,229,238,
%T A039373 265,281,326,335,342,343,345,347,348,349,350,353,371,380,389,398,427,
%U A039373 443,508,524,589,605,670,686,751,767,832,848,895,904,922,927,928
%N A039373 Representation in base 9 has same nonzero number of 2's and 4's.
%K A039373 nonn,base,easy
%O A039373 0,1
%A A039373 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043196
%S A043196 22,38,103,119,184,198,265,281,342,362,427,443,508,524,589,605,
%T A043196 670,686,751,767,832,848,913,927,994,1010,1071,1091,1156,1172,
%U A043196 1237,1253,1318,1334,1399,1415,1480,1496,1561,1577,1642,1656
%N A043196 2 and 4 occur juxtaposed in the base 9 representation of n but not of n-1.
%K A043196 nonn,base
%O A043196 1,1
%A A043196 Clark Kimberling, ck6@cedar.evansville.edu
%I A043976
%S A043976 22,38,103,119,184,206,265,281,350,362,427,443,508,524,589,605,
%T A043976 670,686,751,767,832,848,913,935,994,1010,1079,1091,1156,1172,
%U A043976 1237,1253,1318,1334,1399,1415,1480,1496,1561,1577,1642,1664
%N A043976 2 and 4 occur juxtaposed in the base 9 representation of n but not of n+1.
%K A043976 nonn,base
%O A043976 1,1
%A A043976 Clark Kimberling, ck6@cedar.evansville.edu
%I A026061
%S A026061 22,40,63,92,127,170,220,278,344,420,505,600,705,822,950,1090,1242,1408,1587,
%T A026061 1780,1987,2210,2448,2702,2972,3260,3565,3888,4229,4590,4970,5370,5790,6232,
%U A026061 6695,7180,7687,8218,8772,9350,9952,10580,11233,11912,12617,13350,14110,14898
%N A026061 a(n) = (d(n)-r(n))/2, where d = A026060 and r is the periodic sequence with fund. period (1,0,0,0).
%K A026061 nonn
%O A026061 5,1
%A A026061 Clark Kimberling (ck6@cedar.evansville.edu)
%I A063327
%S A063327 1,22,42,64,86,106,128,150,170,192,214,234,256,278,298,320,342,
%T A063327 362,384,406,426,448,470,490,512,534,554,576,598,618,640,662,682,
%U A063327 704,726,746,768,790,810,832,854,874,896,918,938,960,982,1002,1024
%V A063327 -1,22,42,64,86,106,128,150,170,192,214,234,256,278,298,320,342,
%W A063327 362,384,406,426,448,470,490,512,534,554,576,598,618,640,662,682,
%X A063327 704,726,746,768,790,810,832,854,874,896,918,938,960,982,1002,1024
%N A063327 Dimension of the space of weight n cuspidal newforms for Gamma_1( 54 ).
%H A063327 William A. Stein (was@math.berkeley.edu), Dimensions of the spaces S_k^{new}(Gamma_1(N))
%H A063327 William A. Stein (was@math.berkeley.edu), The modular forms database
%K A063327 sign,done
%O A063327 2,2
%A A063327 njas, Jul 14 2001
%I A066957
%S A066957 22,43,18,41,18,40,24,39,18,46,17,38,27,38,24,43,18,39,23,37,23,43,25,
%T A066957 36,32,36,25,44,25,35,34,31,26,37,24,35,32,32,27,37,31,26,34,34,29,42,
%U A066957 27,27,35,26,28,35,29,33,30,26,23,29,32,25,33,30,24,34,22,26,32,30,25
%N A066957 a(n) = number of ways of placing '+' and '-' among the digits 123456789 so that the result of the expression is n, '-' before 1 IS allowed.
%Y A066957 Cf. A066956.
%K A066957 nice,nonn,new
%O A066957 0,1
%A A066957 Miklos SZABO (mike@ludens.elte.hu), Feb 01 2002
%I A019508
%S A019508 22,43,85,94,105,106,148,169,187,209,211,218,232,274,280,295,313,316,337,
%T A019508 358,373,382,400,417,421,435,463,466,484,521,526,547,559,589,610,625,631,
%U A019508 652,673,715,736,745,763,778,799,833,838,841,862,869,890,904,925,931,937
%N A019508 X^m=X rings without normal forms: integers m>1 for which there exists a prime p and integers a,b>0 such that both p^a-1 and p^b-1 divide m-1 but p^lcm(a,b)-1 does not divide m-1.
%D A019508 Burris, Stanley; Lawrence, John; Term rewrite rules for finite fields Internat. J. Algebra Comput. 1 (1991), no. 3, 353-369. MR 93a:68069 (Reviewer: Ralph W. Wilkerson).
%t A019508 Select[ Range[ 2,1000 ],Function[ m,Module[ {k},Length[ k=Flatten[ Select[ Map[ FactorInteger,1+Divisors[ m-1 ] ],Length[ #1 ]==1&[ [ 1,2 ] ]>1& ],1 ] ]>1&&Length[ Select[ Map[ Function[ p,{p,Last[ Transpose[ Select[ k,#1[ [ 1 ] ]==p& ] ] ]} ],Union[ First[ Transpose[ k ] ] ] ],Length[ #1[ [ 2 ] ] ]>1&&!MemberQ[ #1[ [ 2 ] ],LCM@@#1[ [ 2 ] ] ]& ] ]>0 ] ] ]
%K A019508 nonn
%O A019508 0,1
%A A019508 Bill Dubuque (wgd@martigny.ai.mit.edu)
%E A019508 Mma program 08/97 (oprg).
%I A040462
%S A040462 22,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,
%T A040462 44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,
%U A040462 44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44,44
%N A040462 Continued fraction for sqrt(485).
%H A040462 Index entries for continued fractions for constants
%p A040462 with(numtheory): Digits:=300: convert(evalf(sqrt(485)),confrac);
%K A040462 nonn,cofr,easy
%O A040462 0,1
%A A040462 njas
%I A015799
%S A015799 22,44,60,84,120,132,135,140,156,204,228,231,270,276,348,372,375,
%T A015799 444,462,492,496,516,525,532,564,636,660,700,708,726,732,750,756,
%U A015799 804,852,876,920,935,948,996,1050,1068,1164,1212,1236,1248,1284
%N A015799 phi(n) + 8 | sigma(n).
%K A015799 nonn
%O A015799 0,1
%A A015799 Robert G. Wilson v (rgwv@kspaint.com)
%I A008604
%S A008604 0,22,44,66,88,110,132,154,176,198,220,242,264,286,308,
%T A008604 330,352,374,396,418,440,462,484,506,528,550,572,594,
%U A008604 616,638,660,682,704,726,748,770,792,814,836,858,880
%N A008604 Multiples of 22.
%H A008604 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 334
%K A008604 nonn
%O A008604 0,2
%A A008604 njas
%I A061832
%S A061832 0,22,44,66,88,220,242,264,286,440,462,484,660,682,880,2002,2024,2046,
%T A061832 2068,2200,2222,2244,2266,2288,2420,2442,2464,2486,2640,2662,2684,2860,
%U A061832 2882,4004,4026,4048,4202,4224,4246,4268,4400,4422,4444,4466,4488,4620
%N A061832 Multiples of 11 having only even digits.
%e A061832 264 = 11*24 is a term having all even digits.
%Y A061832 Cf. A061829, A061830, A061831.
%K A061832 nonn,base,easy
%O A061832 0,2
%A A061832 Amarnath Murthy (amarnath_murthy@yahoo.com), May 29 2001
%E A061832 More terms from Larry Reeves (larryr@acm.org), May 30 2001
%I A038153
%S A038153 22,44,67,89,112,134,157,179,202,224,247,269,291,314,336,359,381,404,
%T A038153 426,449,471,494,516,539,561,583,606,628,651
%N A038153 Beatty sequence for pi^e.
%H A038153 Index entries for sequences related to Beatty sequences
%F A038153 a(n)=int(n*22.4591577....)
%K A038153 nonn
%O A038153 1,1
%A A038153 Felice Russo (felice.russo@katamail.com)
%I A033848
%S A033848 22,44,88,176,242,352,484,704,968,1408,1936,2662,2816,3872,5324,5632,
%T A033848 7744,10648,11264,15488,21296,22528,29282,30976,42592,45056,58564,
%U A033848 61952,85184,90112,117128,123904,170368,180224,234256,247808,322102
%N A033848 Prime factors are 2 and 11.
%K A033848 nonn
%O A033848 0,1
%A A033848 Jeff Burch (gburch@erols.com)
%I A003858
%S A003858 1,22,45,45,230,231,231,231,253,770,770,896,896,990,990,
%T A003858 1035,2024
%N A003858 Degrees of irreducible representations of group M23.
%D A003858 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%o A003858 (GAP) Display(CharacterTable("M23"));
%K A003858 nonn,fini,full
%O A003858 1,2
%A A003858 njas
%I A041956
%S A041956 22,45,67,112,179,291,1052,8707,9759,28225,292009,320234,
%T A041956 1893179,4106592,59385467,122877526,673773097,796650623,
%U A041956 8640279327,18077209277,26717488604,231817118109,722168842931
%N A041956 Numerators of continued fraction convergents to sqrt(501).
%Y A041956 Cf. A041957.
%K A041956 nonn,cofr,easy
%O A041956 0,1
%A A041956 njas
%I A041954
%S A041954 22,45,67,246,559,805,1364,14445,15809,30254,76317,259205,
%T A041954 335522,930249,41266478,83463205,124729683,457652254,1040034191,
%U A041954 1497686445,2537720636,26874892805,29412613441,56287506246
%N A041954 Numerators of continued fraction convergents to sqrt(500).
%Y A041954 Cf. A041955.
%K A041954 nonn,cofr,easy
%O A041954 0,1
%A A041954 njas
%I A041952
%S A041952 22,45,67,1452,1519,4490,199079,402648,601727,13038915,
%T A041952 13640642,40320199,1787729398,3615778995,5403508393,117089455248,
%U A041952 122492963641,362075382530,16053809794961,32469694972452
%N A041952 Numerators of continued fraction convergents to sqrt(499).
%Y A041952 Cf. A041953.
%K A041952 nonn,cofr,easy
%O A041952 0,1
%A A041952 njas
%I A041960
%S A041960 22,45,112,157,3409,3566,10541,24648,1095053,2214754,5524561,
%T A041960 7739315,168050176,175789491,519629158,1215047807,53981732666,
%U A041960 109178513139,272338758944,381517272083,8284201472687,8665718744770
%N A041960 Numerators of continued fraction convergents to sqrt(503).
%Y A041960 Cf. A041961.
%K A041960 nonn,cofr,easy
%O A041960 0,1
%A A041960 njas
%I A041958
%S A041958 22,45,112,829,11718,12547,61906,1374479,5559822,6934301,
%T A041958 102640036,725414553,1553469142,3832352837,170176993970,
%U A041958 344186340777,858549675524,6354034069445,89815026647754
%N A041958 Numerators of continued fraction convergents to sqrt(502).
%Y A041958 Cf. A041959.
%K A041958 nonn,cofr,easy
%O A041958 0,1
%A A041958 njas
%I A041962
%S A041962 22,45,202,449,19958,40365,181418,403201,17922262,36247725,
%T A041962 162913162,362074049,16094171318,32550416685,146295838058,
%U A041962 325142092801,14452547921302,29230237935405,131373499662922
%N A041962 Numerators of continued fraction convergents to sqrt(504).
%Y A041962 Cf. A041963.
%K A041962 nonn,cofr,easy
%O A041962 0,1
%A A041962 njas
%I A041964
%S A041964 22,45,382,809,35978,72765,618098,1308961,58212382,117733725,
%T A041964 1000082182,2117898089,94187598098,190493094285,1618132352378,
%U A041964 3426757799041,152395475510182,308217708819405,2618137146065422
%N A041964 Numerators of continued fraction convergents to sqrt(505).
%Y A041964 Cf. A041965.
%K A041964 nonn,cofr,easy
%O A041964 0,1
%A A041964 njas
%I A041966
%S A041966 22,45,2002,4049,180158,364365,16212218,32788801,1458919462,
%T A041966 2950627725,131286539362,265523706449,11814329623118,23894182952685,
%U A041966 1063158379541258,2150210942035201,95672439829090102,193495090600215405
%N A041966 Numerators of continued fraction convergents to sqrt(506).
%Y A041966 Cf. A041967.
%K A041966 nonn,cofr,easy
%O A041966 0,1
%A A041966 njas
%I A044099
%S A044099 22,47,72,97,110,122,147,172,197,222,235,247,272,297,322,347,
%T A044099 360,372,397,422,447,472,485,497,522,547,550,597,610,622,647,
%U A044099 672,697,722,735,747,772,797,822,847,860,872,897,922,947,972
%N A044099 String 4,2 occurs in the base 5 representation of n but not of n-1.
%K A044099 nonn,base
%O A044099 1,1
%A A044099 Clark Kimberling, ck6@cedar.evansville.edu
%I A044480
%S A044480 22,47,72,97,114,122,147,172,197,222,239,247,272,297,322,347,364,372,
%T A044480 397,422,447,472,489,497,522,547,574,597,614,622,647,672,697,722,739,
%U A044480 747,772,797,822,847,864,872,897,922,947,972
%N A044480 String 4,2 occurs in the base 5 representation of n but not of n+1.
%K A044480 nonn,base
%O A044480 1,1
%A A044480 Clark Kimberling, ck6@cedar.evansville.edu
%I A039345
%S A039345 22,50,86,114,134,142,158,166,174,176,177,179,180,181,183,190,214,242,
%T A039345 278,306,342,370,386,394,400,401,403,404,405,407,410,418,426,442,470,
%U A039345 498,534,562,598,626,646,654,670,678,686,688,689,691,692,693,695,702
%N A039345 Representation in base 8 has same nonzero number of 2's and 6's.
%K A039345 nonn,base,easy
%O A039345 0,1
%A A039345 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043168
%S A043168 22,50,86,114,150,176,214,242,278,306,342,370,400,434,470,498,
%T A043168 534,562,598,626,662,688,726,754,790,818,854,882,912,946,982,
%U A043168 1010,1046,1074,1110,1138,1174,1200,1238,1266,1302,1330,1366
%N A043168 2 and 6 occur juxtaposed in the base 8 representation of n but not of n-1.
%K A043168 nonn,base
%O A043168 1,1
%A A043168 Clark Kimberling, ck6@cedar.evansville.edu
%I A043948
%S A043948 22,50,86,114,150,183,214,242,278,306,342,370,407,434,470,498,
%T A043948 534,562,598,626,662,695,726,754,790,818,854,882,919,946,982,
%U A043948 1010,1046,1074,1110,1138,1174,1207,1238,1266,1302,1330,1366
%N A043948 2 and 6 occur juxtaposed in the base 8 representation of n but not of n+1.
%K A043948 nonn,base
%O A043948 1,1
%A A043948 Clark Kimberling, ck6@cedar.evansville.edu
%I A015226
%S A015226 22,50,252,372,946,1222,2360,2856,4750,5530,8372,9500,13482,15022,
%T A015226 20336,22352,29190,31746,40300,43460,53922,57750,70312,74872,89726,
%U A015226 95082,112420,118636,138650,145790,168672,176800,202742,211922,241116
%N A015226 Even hexagonal pyramidal numbers.
%F A015226 Even numbers of form n(n+1)(4n-1)/6.
%t A015226 Select[ Table[ n(n+1)(4n-1)/6,{n,100} ],EvenQ ]
%K A015226 nonn,easy
%O A015226 0,1
%A A015226 Mohammad K. Azarian (ma3@cedar.evansville.edu)
%E A015226 More terms from Erich Friedman (erich.friedman@stetson.edu).
%I A063302
%S A063302 1,22,56,91,126,161,196,231,266,301,336,369,406,441,476,509,546,
%T A063302 579,616,649,686,719,756,787,826,859,896,927,966,997,1036,1067,
%U A063302 1106,1137,1176,1205,1246,1277,1316,1345,1386,1415,1456,1485,1526
%V A063302 -1,22,56,91,126,161,196,231,266,301,336,369,406,441,476,509,546,
%W A063302 579,616,649,686,719,756,787,826,859,896,927,966,997,1036,1067,
%X A063302 1106,1137,1176,1205,1246,1277,1316,1345,1386,1415,1456,1485,1526
%N A063302 Dimension of the space of weight n cuspidal newforms for Gamma_1( 29 ).
%H A063302 William A. Stein (was@math.berkeley.edu), Dimensions of the spaces S_k^{new}(Gamma_1(N))
%H A063302 William A. Stein (was@math.berkeley.edu), The modular forms database
%K A063302 sign,done
%O A063302 2,2
%A A063302 njas, Jul 14 2001
%I A058097
%S A058097 1,0,22,56,177,352,870,1584,3412,5952,11442,19240,34377,56256,95560,
%T A058097 151824,247965,385024,609756,927864,1431094,2139680,3228516,4752896,
%U A058097 7038610
%N A058097 McKay-Thompson series of class 10A for Monster.
%D A058097 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058097 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K A058097 nonn
%O A058097 -1,3
%A A058097 njas, Nov 27 2000
%I A019506
%S A019506 22,58,84,85,94,136,160,166,202,234,250,265,274,308,319,336,346,355,
%T A019506 361,364,382,391,424,438,454,456,476,483,516,517,526,535,562,627,
%U A019506 634,644,645,650,654,660,663,690,702,706,732,735,762,778,855,860
%N A019506 Hoax numbers: composite numbers whose digit-sum equals the sum of the digit-sums of its distinct prime factors.
%H A019506 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A019506 Cf. A006753.
%K A019506 nonn,base
%O A019506 1,1
%A A019506 Mario Velucchi (velucchi@CLI.DI.Unipi.IT)
%I A044124
%S A044124 22,58,94,130,132,166,202,238,274,310,346,348,382,418,454,490,
%T A044124 526,562,564,598,634,670,706,742,778,780,792,850,886,922,958,
%U A044124 994,996,1030,1066,1102,1138,1174,1210,1212,1246,1282,1318,1354
%N A044124 String 3,4 occurs in the base 6 representation of n but not of n-1.
%K A044124 nonn,base
%O A044124 1,1
%A A044124 Clark Kimberling, ck6@cedar.evansville.edu
%I A044505
%S A044505 22,58,94,130,137,166,202,238,274,310,346,353,382,418,454,490,526,562,
%T A044505 569,598,634,670,706,742,778,785,827,850,886,922,958,994,1001,1030,
%U A044505 1066,1102,1138,1174,1210,1217,1246,1282,1318,1354
%N A044505 String 3,4 occurs in the base 6 representation of n but not of n+1.
%K A044505 nonn,base
%O A044505 1,1
%A A044505 Clark Kimberling, ck6@cedar.evansville.edu
%I A051874
%S A051874 0,1,22,63,124,205,306,427,568,729,910,1111,1332,1573,1834,2115,
%T A051874 2416,2737,3078,3439,3820,4221,4642,5083,5544,6025,6526,7047,7588,
%U A051874 8149,8730,9331,9952,10593,11254,11935,12636,13357,14098,14859
%N A051874 22-gonal numbers.
%D A051874 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
%F A051874 a(n)=n(10n-9).
%K A051874 nonn
%O A051874 0,3
%A A051874 njas, Dec 15 1999
%I A064710
%S A064710 1,22,66,70,81,94,115,119,170,210,214,217,265,282,310,322,343,345,357,
%T A064710 364,382,385,472,497,510,517,527,642,651,679,710,742,745,782,795,
%U A064710 820,862,884,889,930,935,966,970,1029,1066,1080,1092,1146,1155,1174
%N A064710 Sum of divisors of n and product of divisors of n are both perfect squares.
%o A064710 (PARI) a(n) = n^(numdiv(n)/2) for(n=1,2000,if(issquare(sigma(n)) && issquare(a(n)),print1(n,",")))
%Y A064710 Cf. A007955, A000203.
%K A064710 easy,nonn
%O A064710 1,2
%A A064710 Jason Earls (jcearls@kskc.net), Oct 13 2001
%E A064710 Corrected by Harvey P. Dale (hpd1@nyu.edu), Oct 23 2001
%I A061595
%S A061595 22,66,112,114,121,123,129,132,141,147,156,165,174,189,192,198,211,213,
%T A061595 219,231,237,273,279,291,297,312,321,327,345,354,369,372,396,411,417,
%U A061595 435,453,459,468,471,477,486,495,516,534,543,549,561,567,576,594,615
%N A061595 Product of digits + 1 is prime, sum of digits + 1 is prime and sum of digits - 1 is prime.
%e A061595 For 147 we have (1*4*7)+1=29, (1+4+7)+1=13, (1+4+7)-1=11.
%K A061595 nonn,base
%O A061595 0,1
%A A061595 Felice Russo (felice.russo@katamail.com), May 22 2001
%I A041946
%S A041946 22,67,89,245,1069,1314,2383,6080,14543,35166,49709,84875,
%T A041946 389209,863293,1252502,4620799,204567658,618323773,822891431,
%U A041946 2264106635,9879317971,12143424606,22022742577,56188909760
%N A041946 Numerators of continued fraction convergents to sqrt(496).
%Y A041946 Cf. A041947.
%K A041946 nonn,cofr,easy
%O A041946 0,1
%A A041946 njas
%I A041948
%S A041948 22,67,156,379,2051,12685,65476,143637,352750,1201887,53235778,
%T A041948 160909221,375054220,911017661,4930142525,30491872811,157389506580,
%U A041948 345270885971,847931278522,2889064721537,127966779026150
%N A041948 Numerators of continued fraction convergents to sqrt(497).
%Y A041948 Cf. A041949.
%K A041948 nonn,cofr,easy
%O A041948 0,1
%A A041948 njas
%I A002757 M5123 N2219
%S A002757 22,67,181,401,831,1576,2876,4987,8406,13715,21893,34134,52327,
%T A002757 78785,116982,171259,247826,354482,502090,704265,979528,1351109,1849932
%N A002757 Bipartite partitions.
%D A002757 M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.
%K A002757 nonn
%O A002757 0,1
%A A002757 njas
%I A041950
%S A041950 22,67,424,9395,56794,179777,7966982,24080723,152451320,
%T A041950 3378009763,20420509898,64639539457,2864560246006,8658320277475,
%U A041950 54814481910856,1214576922316307,7342276015808698,23241404969742401
%N A041950 Numerators of continued fraction convergents to sqrt(498).
%Y A041950 Cf. A041951.
%K A041950 nonn,cofr,easy
%O A041950 0,1
%A A041950 njas
%I A044160
%S A044160 22,71,120,154,169,218,267,316,365,414,463,497,512,561,610,659,
%T A044160 708,757,806,840,855,904,953,1002,1051,1078,1149,1183,1198,1247,
%U A044160 1296,1345,1394,1443,1492,1526,1541,1590,1639,1688,1737,1786
%N A044160 String 3,1 occurs in the base 7 representation of n but not of n-1.
%K A044160 nonn,base
%O A044160 1,1
%A A044160 Clark Kimberling, ck6@cedar.evansville.edu
%I A044541
%S A044541 22,71,120,160,169,218,267,316,365,414,463,503,512,561,610,659,708,757,
%T A044541 806,846,855,904,953,1002,1051,1126,1149,1189,1198,1247,1296,1345,1394,
%U A044541 1443,1492,1532,1541,1590,1639,1688,1737,1786
%N A044541 String 3,1 occurs in the base 7 representation of n but not of n+1.
%K A044541 nonn,base
%O A044541 1,1
%A A044541 Clark Kimberling, ck6@cedar.evansville.edu
%I A003908
%S A003908 1,22,77,154,154,154,175,231,693,770,770,770,825,896,896,
%T A003908 1056,1386,1408,1750,1925,1925,2520,2750,3200
%N A003908 Degrees of irreducible representations of group HS.
%D A003908 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%o A003908 (GAP) Display(CharacterTable("HS"));
%K A003908 nonn,fini,full
%O A003908 1,2
%A A003908 njas
%I A010010
%S A010010 1,22,82,182,322,502,722,982,1282,1622,2002,2422,2882,3382,
%T A010010 3922,4502,5122,5782,6482,7222,8002,8822,9682,10582,11522,
%U A010010 12502,13522,14582,15682,16822,18002,19222,20482,21782
%N A010010 a(0)=1, a(n)=20*n^2 + 2, n >= 1.
%K A010010 nonn
%O A010010 0,2
%A A010010 njas
%I A044209
%S A044209 22,86,150,176,214,278,342,406,470,534,598,662,688,726,790,854,
%T A044209 918,982,1046,1110,1174,1200,1238,1302,1366,1408,1494,1558,1622,
%U A044209 1686,1712,1750,1814,1878,1942,2006,2070,2134,2198,2224,2262
%N A044209 String 2,6 occurs in the base 8 representation of n but not of n-1.
%K A044209 nonn,base
%O A044209 1,1
%A A044209 Clark Kimberling, ck6@cedar.evansville.edu
%I A044590
%S A044590 22,86,150,183,214,278,342,406,470,534,598,662,695,726,790,854,918,982,
%T A044590 1046,1110,1174,1207,1238,1302,1366,1471,1494,1558,1622,1686,1719,1750,
%U A044590 1814,1878,1942,2006,2070,2134,2198,2231,2262
%N A044590 String 2,6 occurs in the base 8 representation of n but not of n+1.
%K A044590 nonn,base
%O A044590 1,1
%A A044590 Clark Kimberling, ck6@cedar.evansville.edu
%I A041940
%S A041940 22,89,111,1199,3708,12323,126938,139261,683982,30234469,
%T A041940 121621858,151856327,1640185128,5072411711,16857420261,
%U A041940 173646614321,190504034582,935662752649,41359665151138
%N A041940 Numerators of continued fraction convergents to sqrt(493).
%Y A041940 Cf. A041941.
%K A041940 nonn,cofr,easy
%O A041940 0,1
%A A041940 njas
%I A041942
%S A041942 22,89,200,489,689,1867,2556,6979,16514,73035,3230054,12993251,
%T A041942 29216556,71426363,100642919,272712201,373355120,1019422441,
%U A041942 2412200002,10668222449,471813987758,1897924173481,4267662334720
%N A041942 Numerators of continued fraction convergents to sqrt(494).
%Y A041942 Cf. A041943.
%K A041942 nonn,cofr,easy
%O A041942 0,1
%A A041942 njas
%I A041944
%S A041944 22,89,3938,15841,700942,2819609,124763738,501874561,22207244422,
%T A041944 89330852249,3952764743378,15900389825761,703569917076862,
%U A041944 2830180058133209,125231492474938058,503756149957885441
%N A041944 Numerators of continued fraction convergents to sqrt(495).
%Y A041944 Cf. A041945.
%K A041944 nonn,cofr,easy
%O A041944 0,1
%A A041944 njas
%I A026909
%S A026909 22,99,379,1412,5265,19758,74637,283560,1082449,4148603,15953607,
%T A026909 61526969,237876571,921678876,3577968081,13913243136,54184698801,
%U A026909 211307360871,825059443551,3225071709981,12619275028611,49423455006501
%N A026909 (1/2)*T(2n,n), T given by A026907.
%K A026909 nonn
%O A026909 1,1
%A A026909 Clark Kimberling, ck6@cedar.evansville.edu
%I A060382
%S A060382 22,100,255,708,1079,2656,1021,593,196,1011,237,2196,361,447,413,3297,
%T A060382 519,341
%N A060382 In base n, a(n) is the smallest number m that leads to a palindrome-free sequence, using the following process: start with m; reverse the digits and add it to m, repeat. Stop if you reach a palindrome.
%H A060382 K. S. Brown, Digit Reversal Sums Leading to Palindromes
%K A060382 nonn
%O A060382 2,1
%A A060382 Michel ten Voorde (upquark@gmx.net), Apr 03 2001
%I A044273
%S A044273 22,103,184,198,265,346,427,508,589,670,751,832,913,927,994,1075,
%T A044273 1156,1237,1318,1399,1480,1561,1642,1656,1723,1782,1885,1966,
%U A044273 2047,2128,2209,2290,2371,2385,2452,2533,2614,2695,2776,2857
%N A044273 String 2,4 occurs in the base 9 representation of n but not of n-1.
%K A044273 nonn,base
%O A044273 1,1
%A A044273 Clark Kimberling, ck6@cedar.evansville.edu
%I A044654
%S A044654 22,103,184,206,265,346,427,508,589,670,751,832,913,935,994,1075,1156,
%T A044654 1237,1318,1399,1480,1561,1642,1664,1723,1862,1885,1966,2047,2128,2209,
%U A044654 2290,2371,2393,2452,2533,2614,2695,2776,2857
%N A044654 String 2,4 occurs in the base 9 representation of n but not of n+1.
%K A044654 nonn,base
%O A044654 1,1
%A A044654 Clark Kimberling, ck6@cedar.evansville.edu
%I A066450
%S A066450 22,103,290,708,1079,2656,1021,593,196,1011,237,2701,361,447,413,3297,
%T A066450 519,341,379,711,461,505,551,1022,649,701,755,811,869,929,991,1055,
%U A066450 1799,1922,1259,1331,1405,1481,1559,1639,1595,1762,1891,1934,2069,2161
%N A066450 Conjectured values for the minimal number a(n) so that the 'reverse and add!'-algorithm in base n does not terminate in a palindrome. If there is no such number in base n, then a(n):=-1.
%C A066450 It would be nice to remove the word "Conjectured" from the description - njas
%C A066450 All the terms in this sequence except the first are only conjectures. (See Walker, Irvin on a(10)=196, and Brockhaus on a(2)=22).
%C A066450 An obvious algorithm is: Start with r:=n and check whether the 'reverse and add!'-algorithm in base n halts in a palindrome or not. If it stops, increment r by one and repeat the process, else return r. To obtain the values above, an upper limit of 100 'reverse and add!'-steps was used.
%C A066450 Conjectures: a(n) shows the same asymptotic behaviour as n^2. For infinitely many n, a(n)=n^2-n-1. Again, it is an open question, if the values of the sequence really lead to infinitely many 'reverse and add!' steps or not. Is the sequence always positive?
%H A066450 K. Brockhaus, On the 'Reverse and Add!' algorithm in base 2
%H A066450 T. Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing.
%H A066450 J. Walker, Three Years Of Computing: Final Report On The Palindrome Quest
%K A066450 nonn
%O A066450 2,1
%A A066450 Frederick Magata (frederick.magata@t-online.de), Dec 29 2001
%E A066450 David W. Wilson remarks (Jan 02, 2002): I verified these using 1000 digits as a stopping point (this would be >>1000 iterations). I am highly confident of these values.
%I A041938
%S A041938 22,111,133,244,2573,2817,5390,29767,1315138,6605457,7920595,
%T A041938 14526052,153181115,167707167,320888282,1772148577,78295425670,
%U A041938 393249276927,471544702597,864793979524,9119484497837,9984278477361
%N A041938 Numerators of continued fraction convergents to sqrt(492).
%Y A041938 Cf. A041939.
%K A041938 nonn,cofr,easy
%O A041938 0,1
%A A041938 njas
%I A061596
%S A061596 22,114,123,129,132,141,156,165,189,192,198,213,219,231,237,273,291,
%T A061596 312,321,327,345,354,372,411,435,453,459,468,486,495,516,534,543,549,
%U A061596 561,594,615,648,651,684,723,732,819,846,864,891,912,918,921,945,954
%N A061596 Product of digits + 1 is prime, product of digits - 1 is prime, sum of digits + 1 is prime and sum of digits - 1 is prime.
%e A061596 22 belong to the sequence because (2*2)+1=5, (2*2)-1=3, (2+2)+1=5, (2+2)-1=3.
%K A061596 base,nonn
%O A061596 0,1
%A A061596 Felice Russo (felice.russo@katamail.com), May 22 2001
%I A039612
%S A039612 22,121,154,178,190,202,214,226,238,250,262,264,266,267,268,269,270,
%T A039612 271,272,273,275,286,310,409,454,553,598,697,742,841,886,985,1030,1129,
%U A039612 1174,1273,1318,1417,1441,1452,1454,1455,1456,1457,1458,1459
%N A039612 Representation in base 12 has same nonzero number of 1's and 10's.
%K A039612 nonn,base,easy
%O A039612 0,1
%A A039612 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043498
%S A043498 22,122,202,212,220,221,223,224,225,226,227,228,229,232,242,252,
%T A043498 262,272,282,292,322,422,522,622,722,822,922,1022,1122,1202,1212,
%U A043498 1220,1221,1223,1224,1225,1226,1227,1228,1229,1232,1242,1252
%N A043498 Number of 2's in base 10 is 2.
%K A043498 nonn,base
%O A043498 1,1
%A A043498 Clark Kimberling, ck6@cedar.evansville.edu
%I A044354
%S A044354 22,122,220,322,422,522,622,722,822,922,1022,1122,1220,1322,1422,
%T A044354 1522,1622,1722,1822,1922,2022,2122,2200,2322,2422,2522,2622,
%U A044354 2722,2822,2922,3022,3122,3220,3322,3422,3522,3622,3722,3822
%N A044354 String 2,2 occurs in the base 10 representation of n but not of n-1.
%K A044354 nonn,base
%O A044354 1,1
%A A044354 Clark Kimberling, ck6@cedar.evansville.edu
%I A044735
%S A044735 22,122,229,322,422,522,622,722,822,922,1022,1122,1229,1322,1422,1522,
%T A044735 1622,1722,1822,1922,2022,2122,2299,2322,2422,2522,2622,2722,2822,2922,
%U A044735 3022,3122,3229,3322,3422,3522,3622,3722,3822
%N A044735 String 2,2 occurs in the base 10 representation of n but not of n+1.
%K A044735 nonn,base
%O A044735 1,1
%A A044735 Clark Kimberling, ck6@cedar.evansville.edu
%I A061597
%S A061597 22,123,132,213,231,312,321,111126,111162,111216,111261,111612,111621,
%T A061597 112116,112161,112611,116112,116121,116211,121116,121161,121611,126111,
%U A061597 161112,161121,161211,162111,211116,211161,211611,216111,261111,611112
%N A061597 Product of digits + 1 is prime, product of digits - 1 is prime, sum of digits + 1 is prime, sum of digits - 1 is prime, and product of digits = sum of digits.
%e A061597 22 belong to the sequence because (2*2)+1=5, (2*2)-1=3, (2+2)+1=5, (2+2)-1=3 and 2+2=2*2.
%K A061597 nonn,base
%O A061597 0,1
%A A061597 Felice Russo (felice.russo@katamail.com), May 22 2001
%I A039439
%S A039439 22,123,143,243,245,246,247,248,249,250,251,252,253,275,286,297,308,
%T A039439 319,330,341,352,365,385,486,506,607,627,728,748,849,869,970,990,1091,
%U A039439 1111,1212,1232,1344,1354,1356,1357,1358,1359,1360,1361,1362,1363
%N A039439 Representation in base 11 has same nonzero number of 0's and 2's.
%K A039439 nonn,base,easy
%O A039439 0,1
%A A039439 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A003778
%S A003778 1,22,132,1006,4324,26996,109722,602804,2434670,12287118,49852352,
%T A003778 237425498,969300694,4434629912,18203944458,80978858522,
%U A003778 333840165288,1456084764388,6021921661718,25904211802080
%N A003778 Hamilton paths in P_5 X P_n.
%H A003778 F. Faase, Counting Hamilton cycles in product graphs
%K A003778 nonn
%O A003778 1,2
%A A003778 Frans Faase (Frans_LiXia@wxs.nl)
%I A041936
%S A041936 22,133,421,1817,14957,16774,31731,48505,128741,177246,
%T A041936 305987,6602973,6908960,13511933,33932826,47444759,81377585,
%U A041936 128822344,1111956337,4576647692,14841899413,93628044170
%N A041936 Numerators of continued fraction convergents to sqrt(491).
%Y A041936 Cf. A041937.
%K A041936 nonn,cofr,easy
%O A041936 0,1
%A A041936 njas
%I A041934
%S A041934 22,155,332,487,2280,9607,40708,50315,141338,1039681,45887302,
%T A041934 322250795,690388892,1012639687,4740947640,19976430247,
%U A041934 84646668628,104623098875,293892866378,2161873163521,95416312061302
%N A041934 Numerators of continued fraction convergents to sqrt(490).
%Y A041934 Cf. A041935.
%K A041934 nonn,cofr,easy
%O A041934 0,1
%A A041934 njas
%I A027943
%S A027943 1,22,155,709,2587,8273,24416,68595,187030,500950,1327986,3499982,
%T A027943 9195035,24115804,63192397,165512723,433410661,1134800215,2971089810,
%U A027943 7778591025
%N A027943 T(2n+1,n+3), T given by A027935.
%K A027943 nonn
%O A027943 2,2
%A A027943 Clark Kimberling, ck6@cedar.evansville.edu
%I A067561
%S A067561 1,22,167,334,4821,328663,657326
%N A067561 Radii n of circles with integer radius that can approximately be squared integrally: the Floor or Ceiling of Pi*n^2 is an integer square.
%C A067561 Ceiling[Pi*22^2] = 39^2, so 22 is a term of the sequence.
%t A067561 Do[m = Pi*i^2; r = IntegerQ[Sqrt[Floor[m]]]; s = IntegerQ[Sqrt[Ceiling[m]]]; If[r || s, Print[i]], {i, 1, 10^6}]
%K A067561 nonn,uned,new
%O A067561 1,2
%A A067561 Joseph L. Pe (joseph_l_pe@hotmail.com), Jan 29 2002
%I A041932
%S A041932 22,177,199,973,1172,2145,3317,12096,27509,39605,106719,
%T A041932 1533671,3174061,4707732,12589525,42476307,55065832,97542139,
%U A041932 152607971,707974023,860581994,7592629975,334936300894
%N A041932 Numerators of continued fraction convergents to sqrt(489).
%Y A041932 Cf. A041933.
%K A041932 nonn,cofr,easy
%O A041932 0,1
%A A041932 njas
%I A022682
%S A022682 1,22,187,638,561,10582,20460,44132,157311,154132,468666,
%T A022682 1959718,2247421,12556104,8229859,41049558,43660639,121417780,
%U A022682 408706870,100429384,1145215709,2659879552,853739235,13377528824
%V A022682 1,-22,187,-638,-561,10582,-20460,-44132,157311,154132,-468666,
%W A022682 -1959718,2247421,12556104,-8229859,-41049558,-43660639,121417780,
%X A022682 408706870,-100429384,-1145215709,-2659879552,853739235,13377528824
%N A022682 Expansion of Product (1-m*q^m)^22; m=1..inf.
%K A022682 sign,done
%O A022682 0,2
%A A022682 njas
%I A020923
%S A020923 1,22,198,924,2310,2772,924,264,198,220,308,504,924,1848,
%T A020923 3960,8976,21318,52668,134596,354200,956340,2641320,7443720,
%U A020923 21360240,62300700,184410072,553230216,1680180656,5160554872
%V A020923 1,-22,198,-924,2310,-2772,924,264,198,220,308,504,924,1848,
%W A020923 3960,8976,21318,52668,134596,354200,956340,2641320,7443720,
%X A020923 21360240,62300700,184410072,553230216,1680180656,5160554872
%N A020923 Expansion of (1-4*x)^(11/2).
%K A020923 sign,done
%O A020923 0,2
%A A020923 njas
%I A031202
%S A031202 22,202,220,256,265,526,562,577,625,652,757,775,1123,1132,1213,1231,
%T A031202 1237,1273,1312,1321,1327,1369,1372,1396,1456,1465,1546,1564,1639,
%U A031202 1645,1654,1693,1723,1732,1789,1798,1879,1897,1936,1963,1978,1987
%N A031202 Numbers having period-2 7-digitized sequences.
%H A031202 E. W. Weisstein, Link to a section of The World of Mathematics.
%K A031202 nonn
%O A031202 0,1
%A A031202 Eric W. Weisstein (eric@weisstein.com)
%I A010828
%S A010828 1,22,209,1078,2926,1672,15169,47234,31350,107426,218680,
%T A010828 266,234707,237006,405878,1444806,2415413,1091398,3018169,
%U A010828 523050,1618309,7344304,134905,5365866,5852,17297588,24278276
%V A010828 1,-22,209,-1078,2926,-1672,-15169,47234,-31350,-107426,218680,
%W A010828 -266,-234707,-237006,405878,1444806,-2415413,-1091398,3018169,
%X A010828 523050,1618309,-7344304,-134905,5365866,5852,17297588,-24278276
%N A010828 Expansion of Product (1-x^k )^22.
%D A010828 Newman, Morris; A table of the coefficients of the powers of $\eta(\tau)$. Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
%K A010828 sign,done
%O A010828 0,2
%A A010828 njas
%I A022714
%S A022714 1,22,209,1122,3894,10956,35123,120362,337106,821854,2257728,
%T A022714 6472050,15946777,37180990,95117858,239904786,552195127,
%U A022714 1268645554,3023055717,7022547026,15717464257,35081020348
%V A022714 1,-22,209,-1122,3894,-10956,35123,-120362,337106,-821854,2257728,
%W A022714 -6472050,15946777,-37180990,95117858,-239904786,552195127,
%X A022714 -1268645554,3023055717,-7022547026,15717464257,-35081020348
%N A022714 Expansion of Product (1+m*q^m)^-22; m=1..inf.
%K A022714 sign,done
%O A022714 0,2
%A A022714 njas
%I A008453
%S A008453 1,22,220,1320,5302,15224,33528,63360,116380,209550,339064,
%T A008453 491768,719400,1095160,1538416,1964160,2624182,3696880,
%U A008453 4763220,5686648,7217144,9528816,11676280,13495680,16317048
%N A008453 Number of ways of writing n as a sum of 11 squares.
%D A008453 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
%D A008453 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.
%H A008453 Index entries for sequences related to sums of squares
%p A008453 (sum(x^(m^2),m=-10..10))^11;
%K A008453 nonn
%O A008453 0,2
%A A008453 njas
%I A066573
%S A066573 22,223,22,421,283,355
%N A066573 f-amicable numbers where f(n) = Floor(|n sin(n)|); f-amicable numbers are defined in A066511.
%H A066573 Pe, J. On a Generalization of Perfect Numbers
%e A066573 Proper divisors of 22 are {1,2,11}; f applied to these = {0, 1, 10}, which sum to 11 = f(223). Proper divisors of 223 are {1}; f applied to these = {0}, which sum to 0 = f(22). Hence, 22, 223 is an f-amicable pair.
%p A066573 f[x_] := Floor[Abs[x*Sin[x]]]; d[x_] := Apply[ Plus, Map[ f, Divisors[ x] ] ] - f[ x]; m = Table[{x, y}, {x, 1, 1000}, {y, 1, 1000}]; Do[a = m[[i, j]]; If[ (a[[1]] < a[[2]]) && (f[a[[1]]] == d[a[[2]]]) && (f[a[[2]]] == d[a[[1]]]), Print[{i, j}]], {i, 1, 1000}, {j, 1, 1000}]
%Y A066573 Cf. A066511, A066218.
%K A066573 nonn
%O A066573 1,1
%A A066573 Joseph L. Pe (joseph_l_pe@hotmail.com), Jan 07 2002
%I A008948
%S A008948 1,22,231,252,385,440,560,616,770,770,1155,1155,1155,1386,
%T A008948 1540,3080,3080,3520,4620,4928,5544,6160,6160,6930,8064,
%U A008948 9240,9240,9240,10395,10395,10395,10395,10395,11264,13860
%N A008948 Degrees of irreducible representations of group U6(2).
%D A008948 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%o A008948 (GAP) Display(CharacterTable("U6(2)"));
%K A008948 nonn,fini
%O A008948 0,2
%A A008948 njas
%I A003909
%S A003909 1,22,231,252,770,770,896,896,1750,3520,3520,4500,4752,
%T A003909 5103,5544,8019,8019,8250,8250,9625,9856,9856,10395,10395
%N A003909 Degrees of irreducible representations of group McL.
%D A003909 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%o A003909 (GAP) Display(CharacterTable("McL"));
%K A003909 nonn,fini,full
%O A003909 1,2
%A A003909 njas
%I A047647
%S A047647 1,22,231,1540,7293,25872,69971,140822,183711,25102,634480,2027804,
%T A047647 3817814,4439116,919600,9829270,27660479,44779042,43632974,1898820,
%U A047647 92518261,219961214,313463842,267448104,15757973,547042056,1173033400
%V A047647 1,-22,231,-1540,7293,-25872,69971,-140822,183711,-25102,-634480,2027804,
%W A047647 -3817814,4439116,-919600,-9829270,27660479,-44779042,43632974,-1898820,
%X A047647 -92518261,219961214,-313463842,267448104,15757973,-547042056,1173033400
%N A047647 Expand {Product_{j=1..inf} (1-x^j) - 1 }^22 in powers of x.
%D A047647 H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
%K A047647 sign,done
%O A047647 1,2
%A A047647 njas
%I A010938
%S A010938 1,22,231,1540,7315,26334,74613,170544,319770,497420,646646,
%T A010938 705432,646646,497420,319770,170544,74613,26334,7315,1540,
%U A010938 231,22,1
%N A010938 Binomial coefficient C(22,n).
%K A010938 nonn,fini,full
%O A010938 0,2
%A A010938 njas
%I A022617
%S A022617 1,22,231,1562,7799,31438,109208,341660,987327,2672868,
%T A022617 6848490,16752958,39388481,89439944,196910681,421739450,
%U A022617 881199561,1800336692,3603535166,7078509064,13665905671
%V A022617 1,-22,231,-1562,7799,-31438,109208,-341660,987327,-2672868,
%W A022617 6848490,-16752958,39388481,-89439944,196910681,-421739450,
%X A022617 881199561,-1800336692,3603535166,-7078509064,13665905671
%N A022617 Expansion of Product (1+q^m)^-22; m=1..inf.
%K A022617 sign,done
%O A022617 0,2
%A A022617 njas
%I A003205 M5124
%S A003205 1,22,234,2348,22726,214642,1993002
%N A003205 Cluster series for f.c.c. lattice.
%D A003205 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 A003205 M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
%H A003205 Index entries for sequences related to f.c.c. lattice
%K A003205 nonn
%O A003205 0,2
%A A003205 njas
%I A037268
%S A037268 1,22,236,244,263,326,333,362,424,442,623,632,2488,2666,2848,2884,
%T A037268 3366,3446,3464,3636,3644,3663,4288,4346,4364,4436,4444,4463,4828,
%U A037268 4882,6266,6336,6344,6363,6434,6443,6626,6633,6662,8248,8284,8428
%N A037268 Sum of reciprocals of digits = 1.
%Y A037268 Cf. A020473, A037264, A038034.
%K A037268 easy,nonn,base,fini
%O A037268 1,2
%A A037268 njas
%E A037268 More terms from Christian G. Bower (bowerc@usa.net), Jun 1998. Sequence contains 1209 terms.
%I A046499
%S A046499 22,242,464,868,4664,8668,40604,44644,48684,80608,84648,88688,406604,
%T A046499 446644,486684,806608,846648,886688,4006004,4046404,4086804,4406044,
%U A046499 4446444,4486844,4806084,4846484,4886884,8006008,8046408,8086808
%N A046499 Palindromes expressible as the sum of 4 consecutive palindromes.
%H A046499 P. De Geest, World!Of Numbers
%e A046499 E.g. 22 = 4 + 5 + 6 + 7; 242 = 44 + 55 + 66 + 77.
%Y A046499 Cf. A002113.
%K A046499 nonn
%O A046499 0,1
%A A046499 Patrick De Geest (pdg@worldofnumbers.com), Sep 1998.
%I A035706
%S A035706 1,22,242,1782,9922,44726,170610,568150,1690370,4573910,
%T A035706 11414898,26572086,58227906,121023606,240089586,457018518,
%U A035706 838478850,1488341910,2564399090,4300978550,7039035586
%N A035706 Coordination sequence for 11-dimensional cubic lattice.
%D A035706 J. Serra-Sagrista and H. Klock, Counting lattice points in l_1 norm, submitted to IEEE Trans. on IT
%H A035706 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).
%F A035706 ((1+x)/(1-x))^11.
%K A035706 nonn,easy
%O A035706 0,2
%A A035706 Serra-Sagrista, Joan; jserra@ccd.uab.es
%E A035706 Recomputed Nov 25 1998 by njas.
%I A041930
%S A041930 22,243,10714,118097,5206982,57394899,2530582538,27893802817,
%T A041930 1229857906486,13556330774163,597708411969658,6588348862440401,
%U A041930 290485058359347302,3201923990815260723,141175140654230819114
%N A041930 Numerators of continued fraction convergents to sqrt(488).
%Y A041930 Cf. A041931.
%K A041930 nonn,cofr,easy
%O A041930 0,1
%A A041930 njas
%I A028571
%S A028571 0,22,252,2332,20002,26062,29392,63736,68886,2701072,2783872,
%T A028571 2884882,29122192,253080352,289050982,25661316652,237776677732,
%U A028571 2393677763932,215331808133512,218759969957812,225588939885522
%N A028571 Palindromes of form n(n+9).
%H A028571 P. De Geest, Palindromic Quasipronics of the form n(n+x)
%K A028571 nonn
%O A028571 0,2
%A A028571 Patrick De Geest (pdg@worldofnumbers.com)
%I A010974
%S A010974 1,22,253,2024,12650,65780,296010,1184040,4292145,14307150,
%T A010974 44352165,129024480,354817320,927983760,2319959400,5567902560,
%U A010974 12875774670,28781143380,62359143990,131282408400,269128937220
%N A010974 Binomial coefficient C(n,21).
%K A010974 nonn
%O A010974 21,2
%A A010974 njas
%I A022587
%S A022587 1,22,253,2046,13134,71368,341275,1473494,5848810,21628002,
%T A022587 75261384,248403586,782547909,2365168542,6887441198,19393122562,
%U A022587 52959869787,140631776582,363943223941,919706094494,2273411319069
%N A022587 Expansion of Product (1+q^m)^22; m=1..inf.
%K A022587 nonn
%O A022587 0,2
%A A022587 njas
%I A004412
%S A004412 1,22,264,2288,15994,95568,505648,2425280,10721832,44229350,
%T A004412 171861360,633713808,2230733648,7532979344,24502989984,
%U A004412 77036477760,234785552122,695409096096,2006117554936,5647472566736
%V A004412 1,-22,264,-2288,15994,-95568,505648,-2425280,10721832,-44229350,
%W A004412 171861360,-633713808,2230733648,-7532979344,24502989984,
%X A004412 -77036477760,234785552122,-695409096096,2006117554936,-5647472566736
%N A004412 Expansion of (Sum x^(n^2), n = -inf .. inf )^(-11).
%K A004412 sign,done
%O A004412 0,2
%A A004412 njas
%I A055756
%S A055756 1,0,0,22,266,0,0,30448,127644,0,0,3702162,9192088,0,0,95482512,
%T A055756 188164638,0,0,1143554434,1960045080,0,0,8506319280,13291819992,0,0,
%U A055756 45759737720,67076102720,0,0,195398075232,272568747924,0,0
%N A055756 Jacobi form of weight 12 and index 1 for Niemeier lattice of type A_12^2.
%D A055756 Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser,1985.
%F A055756 E_8*E_{4,1}-34*phi_12
%Y A055756 A008694.
%K A055756 nonn
%O A055756 0,4
%A A055756 Kok Seng Chua (chuaks@ihpc.nus.edu.sg), Jul 12 2000
%I A023020
%S A023020 1,22,275,2530,18975,122430,702328,3661900,17627775,79264900,335937954,
%T A023020 1351507830,5191041625,19125838600,67862904725,232671319474,773027485065,
%U A023020 2494957906100,7839428942950,24025993453000,71941861591215
%N A023020 Partitions of n into parts of 22 kinds.
%K A023020 nonn
%O A023020 0,2
%A A023020 dww
%I A022650
%S A022650 1,22,275,2574,19943,134618,816596,4543396,23522279,114532572,
%T A022650 528773278,2329635750,9845310101,40080724376,157739993037,
%U A022650 601928809878,2232820512417,8069089487932,28463636262974
%N A022650 Expansion of Product (1+m*q^m)^22; m=1..inf.
%K A022650 nonn
%O A022650 0,2
%A A022650 njas
%I A004316
%S A004316 1,22,276,2600,20475,142506,906192,5379616,30260340,163011640,
%T A004316 847660528,4280561376,21090682613,101766230790,482320623240,
%U A004316 2250829575120,10363194502115,47153358767970,212327989773900
%N A004316 Binomial coefficient C(2n,n-10).
%D A004316 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.
%K A004316 nonn,easy
%O A004316 10,2
%A A004316 njas
%I A020922
%S A020922 1,22,286,2860,24310,184756,1293292,8498776,53117350,318704100,
%T A020922 1848483780,10418726760,57302997180,308554600200,1630931458200,
%U A020922 8480843582640,43464323361030,219878341708740,1099391708543700
%N A020922 Expansion of 1/(1-4*x)^(11/2).
%F A020922 a(n)=binomial(n+5,5)*A000984(n+5)/A000984(5), A000984: central binomial coefficients - from Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de).
%K A020922 nonn
%O A020922 0,2
%A A020922 njas
%I A022746
%S A022746 1,22,297,3058,26334,198748,1353275,8474202,49475074,272055454,
%T A022746 1420063656,7079791314,33881645721,156287683310,697257244178,
%U A022746 3017396237922,12697675601127,52071958360466,208490926189117
%N A022746 Expansion of Product (1-m*q^m)^-22; m=1..inf.
%K A022746 nonn
%O A022746 0,2
%A A022746 njas
%I A061341
%S A061341 22,303,1006,2272,3003,6001,6006,10006,30003,50015,50024,60001,60006,
%T A061341 60025
%N A061341 Numbers not ending in 0 whose cubes are concatenations of other cubes.
%e A061341 2272^3 = 1_1728_0_27_64_8
%Y A061341 Cf. A009421.
%K A061341 base,nonn
%O A061341 0,1
%A A061341 Erich Friedman (efriedma@stetson.edu), Jun 06 2001
%I A028109
%S A028109 1,22,305,3410,33621,305382,2619625,21554170,171870941
%N A028109 Same as A003468.
%K A028109 dead
%O A028109 0,2
%I A003468 M5125
%S A003468 1,22,305,3410,33621,305382,2619625,21554170,171870941,
%T A003468 1337764142,10216988145,76862115330,571247591461,4203844925302,
%U A003468 30687029023865,222518183370890,1604626924403181,11518132293452862
%N A003468 Minimal covers of an n-set.
%D A003468 Hearne and Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.
%H A003468 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A003468 G.f.: 1 / ( 1 - 4 x ) ( 1 - 5 x ) ( 1 - 6 x ) ( 1 - 7 x ).
%K A003468 nonn
%O A003468 3,2
%A A003468 njas
%I A041928
%S A041928 22,309,331,971,4215,5186,9401,33389,76179,414284,490463,
%T A041928 3357062,3847525,7204587,155143852,162348439,317492291,
%U A041928 2067302185,2384794476,13991274565,30367343606,105093305383
%N A041928 Numerators of continued fraction convergents to sqrt(487).
%Y A041928 Cf. A041929.
%K A041928 nonn,cofr,easy
%O A041928 0,1
%A A041928 njas
%I A028054
%S A028054 1,22,309,3542,36197,344190,3119173,27339334,233997093,
%T A028054 1968524558,16351808837,134557949526,1099534023589,8937744119326,
%U A028054 72366036152901,584189060834918,4705458521506085,37837251535931694
%N A028054 Expansion of 1/((1-3x)(1-5x)(1-6x)(1-8x)).
%K A028054 nonn
%O A028054 0,2
%A A028054 njas
%I A028038
%S A028038 1,22,311,3608,37485,363594,3368947,30231916,265060169,
%T A028038 2283904766,19419219183,163405866624,1363677239653,11304693716338,
%U A028038 93205539717419,765024499566932,6255874320057537,50996811157171110
%N A028038 Expansion of 1/((1-3x)(1-4x)(1-7x)(1-8x)).
%K A028038 nonn
%O A028038 0,2
%A A028038 njas
%I A025992
%S A025992 1,22,313,3666,38493,377286,3529681,31947322,282198565,
%T A025992 2447183310,20920905369,176852694018,1481626607917,12322682753494,
%U A025992 101879323774177,838170485025354,6867569457133749,56077266261254238
%N A025992 Expansion of 1/((1-2x)(1-5x)(1-7x)(1-8x)).
%K A025992 nonn
%O A025992 0,2
%A A025992 njas
%I A028034
%S A028034 1,22,313,3682,39109,391174,3769921,35484394,328932637,
%T A028034 3018821806,27523981849,249859595986,2261664434485,20432811986518,
%U A028034 184363359728497,1662082800576058,14975637634031053,134882161611414910
%N A028034 Expansion of 1/((1-3x)(1-4x)(1-6x)(1-9x)).
%K A028034 nonn
%O A028034 0,2
%A A028034 njas
%I A028231
%S A028231 1,22,313,4366,60817,847078,11798281
%N A028231 From hexagons in a circle problem.
%D A028231 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.
%K A028231 nonn
%O A028231 0,2
%A A028231 njas
%I A025988
%S A025988 1,22,315,3740,40121,405042,3935095,37284280,347419941,
%T A025988 3201202862,29273641475,266321046420,2414445954961,21837387317482,
%U A025988 197190900709455,1778695458106160,16032377546389181,144437380023772902
%N A025988 Expansion of 1/((1-2x)(1-5x)(1-6x)(1-9x)).
%K A025988 nonn
%O A025988 0,2
%A A025988 njas
%I A023949
%S A023949 1,22,317,3782,40533,405534,3869149,35663254,320292005,
%T A023949 2819331086,24426173421,208945416966,1768927681717,14848644708478,
%U A023949 123763757447933,1025496480025718,8455062115190469,69418440837750510
%N A023949 Expansion of 1/((1-x)(1-6x)(1-7x)(1-8x)).
%K A023949 nonn
%O A023949 0,2
%A A023949 njas
%I A025972
%S A025972 1,22,317,3806,41421,424974,4198189,40430302,382529741,
%T A025972 3573890606,33084972141,304212832638,2783085627661,25364095795918,
%U A025972 230491675046573,2089930913053214,18917927670786381,171021229569836910
%N A025972 Expansion of 1/((1-2x)(1-4x)(1-7x)(1-9x)).
%K A025972 nonn
%O A025972 0,2
%A A025972 njas
%I A028029
%S A028029 1,22,317,3830,42381,447582,4608877,46813030,471994061,
%T A028029 4740248942,47508048237,475624522230,4759030936141,47604501583102,
%U A028029 476117036332397,4761534717751430,47617186000380621,476181122450194062
%N A028029 Expansion of 1/((1-3x)(1-4x)(1-5x)(1-10x)).
%K A028029 nonn
%O A028029 0,2
%A A028029 njas
%I A022453
%S A022453 1,22,320,3890,42861,444612,4433080,43016380,409466321,
%T A022453 3843870602,35717758440,329381322270,3020187550381,27573503381992,
%U A022453 250911867832400,2277497319225560,20632700169031041,186641825267602782
%N A022453 Expansion of 1/((1-x)(1-5x)(1-7x)(1-9x)).
%K A022453 nonn
%O A022453 0,2
%A A022453 njas
%I A025968
%S A025968 1,22,320,3920,44016,471072,4904320,50237440,509670656,
%T A025968 5141007872,51677982720,518395637760,5193684791296,51995352604672,
%U A025968 520305091051520,5205162447994880,52064302294695936,520719124195049472
%N A025968 Expansion of 1/((1-2x)(1-4x)(1-6x)(1-10x)).
%K A025968 nonn
%O A025968 0,2
%A A025968 njas
%I A025948
%S A025948 1,22,321,3938,44045,465894,4751017,47229226,460842789,
%T A025948 4433881166,42195474113,398084143314,3729357886333,34737030515638,
%U A025948 322015765266009,2973201855377402,27359296571272277,251037508511614110
%N A025948 Expansion of 1/((1-2x)(1-3x)(1-8x)(1-9x)).
%K A025948 nonn
%O A025948 0,2
%A A025948 njas
%I A021904
%S A021904 1,22,323,3992,44949,478074,4896391,48839164,477813017,
%T A021904 4606751006,43913627979,414851195616,3890711480605,36272898564418,
%U A021904 336508406905487,3108995652410948,28624326520414113,262765886779943910
%N A021904 Expansion of 1/((1-x)(1-4x)(1-8x)(1-9x)).
%K A021904 nonn
%O A021904 0,2
%A A021904 njas
%I A022343
%S A022343 1,22,323,4004,45465,491106,5149327,53020528,539857109,
%T A022343 5458923470,54963556011,551942523132,5533572185233,55422129454714,
%U A022343 554747369555975,5550668292585416,55526041242871437,555377516005134438
%N A022343 Expansion of 1/((1-x)(1-5x)(1-6x)(1-10x)).
%K A022343 nonn
%O A022343 0,2
%A A022343 njas
%I A025945
%S A025945 1,22,323,4016,45873,499386,5280511,54817972,562288925,
%T A025945 5721711710,57909049419,583934547768,5873255438857,58969928883394,
%U A025945 591360924613847,5925240739579004,59333827978882869,593908224257636838
%N A025945 Expansion of 1/((1-2x)(1-3x)(1-7x)(1-10x)).
%K A025945 nonn
%O A025945 0,2
%A A025945 njas
%I A020571
%S A020571 1,22,325,4030,45301,478702,4851925,47752510,460048501,
%T A020571 4362445582,40876539925,379553364190,3499808594101,32098136255662,
%U A020571 293160602826325,2668857246099070,24235419069434101,219645625266148942
%N A020571 Expansion of 1/((1-6x)(1-7x)(1-9x)).
%K A020571 nonn
%O A020571 0,2
%A A020571 njas
%I A021874
%S A021874 1,22,325,4070,46781,511742,5430405,56516790,580744461,
%T A021874 5916830062,59935396885,604729235110,6084941584541,61113049957982,
%U A021874 612976296281765,6142684971387030,61517309500479021,615806336417543502
%N A021874 Expansion of 1/((1-x)(1-4x)(1-7x)(1-10x)).
%K A021874 nonn
%O A021874 0,2
%A A021874 njas
%I A025963
%S A025963 1,22,325,4110,48381,550062,6148165,68149870,752379661,
%T A025963 8290355502,91266902805,1004309278830,11049302357341,121551961591342,
%U A025963 1337120292662245,14708568942522990,161795495573813421
%N A025963 Expansion of 1/((1-2x)(1-4x)(1-5x)(1-11x)).
%K A025963 nonn
%O A025963 0,2
%A A025963 njas
%I A020343
%S A020343 1,22,327,4100,46781,503202,5201827,52278640,514595961,
%T A020343 4986022382,47721236927,452316244380,4253691236341,39747202125562,
%U A020343 369442770633627,3418759065309320,31519177212527921,289676122850156742
%N A020343 Expansion of 1/((1-5x)(1-8x)(1-9x)).
%K A020343 nonn
%O A020343 0,2
%A A020343 njas
%I A025941
%S A025941 1,22,327,4172,49553,567714,6382699,71043064,786493125,
%T A025941 8681598926,95678810591,1053554778276,11595631317817,127591121803258,
%U A025941 1403737417995603,15442522109891408,169876206409453229
%N A025941 Expansion of 1/((1-2x)(1-3x)(1-6x)(1-11x)).
%K A025941 nonn
%O A025941 0,2
%A A025941 njas
%I A047868
%S A047868 1,22,328,4400,58140,785256
%N A047868 Erroneous version of A002539.
%K A047868 dead
%O A047868 1,2
%I A002539 M5126 N2221
%S A002539 1,22,328,4400,58140,785304,11026296,162186912,2507481216,40788301824,
%T A002539 697929436800,12550904017920,236908271543040,4687098165573120
%N A002539 Permutations by descents.
%D A002539 O. J. Munch, Om potensproduktsummer [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19.
%D A002539 I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
%Y A002539 3rd diagonal of A008517.
%K A002539 nonn,nice,easy
%O A002539 1,2
%A A002539 njas, sp,Robert G. Wilson v (rgwv@kspaint.com),mb
%E A002539 corrected
%I A019958
%S A019958 1,22,329,4178,48621,537222,5744929,60136378,620564021,
%T A019958 6341995022,64384199529,650640568578,6554239839421,65878458172822,
%U A019958 661143103694129,6627971208280778,66395645870074821,664768758151070622
%N A019958 Expansion of 1/((1-5x)(1-7x)(1-10x)).
%K A019958 nonn
%O A019958 0,2
%A A019958 njas
%I A021644
%S A021644 1,22,329,4202,49437,554070,6019393,64026754,670939973,
%T A021644 6954808718,71511447657,730821626106,7433874254509,75344008884166,
%U A021644 761482226728721,7679159391907058,77306290980599445,777180486491935614
%N A021644 Expansion of 1/((1-x)(1-3x)(1-8x)(1-10x)).
%K A021644 nonn
%O A021644 0,2
%A A021644 njas
%I A021834
%S A021834 1,22,329,4226,50469,580422,6541681,72922570,808020125,
%T A021834 8923802030,98376704361,1083438632562,11925616635349,131228622530326,
%U A021834 1443796242996449,15883448475593882,174728077773953421
%N A021834 Expansion of 1/((1-x)(1-4x)(1-6x)(1-11x)).
%K A021834 nonn
%O A021834 0,2
%A A021834 njas
%I A019671
%S A019671 1,22,332,4280,50736,571872,6238912,66567040,699159296,
%T A019671 7259766272,74744097792,764616652800,7783588704256,78935331561472,
%U A019671 798149140201472,8051859072450560,81081536382959616,815318946277097472
%N A019671 Expansion of 1/((1-4x)(1-8x)(1-10x)).
%K A019671 nonn
%O A019671 0,2
%A A019671 njas
%I A021614
%S A021614 1,22,332,4322,52353,609924,6948544,78112924,871004585,
%T A021614 9663416906,106874239836,1179653481006,13004446987897,143246730136168,
%U A021614 1577098731557408,17357778969071768,191003419177941489
%N A021614 Expansion of 1/((1-x)(1-3x)(1-7x)(1-11x)).
%K A021614 nonn
%O A021614 0,2
%A A021614 njas
%I A019854
%S A019854 1,22,333,4334,52325,606606,6874477,76908238,854115189,
%T A019854 9445967630,104219612861,1148348383182,12643672205893,139152654913294,
%U A019854 1531118871452685,16844976107996366,185310900907951637
%N A019854 Expansion of 1/((1-5x)(1-6x)(1-11x)).
%K A019854 nonn
%O A019854 0,2
%A A019854 njas
%I A025936
%S A025936 1,22,333,4406,55133,673566,8144701,98052262,1178225565,
%T A025936 14146756910,169801508669,2037820760118,24454863987997,
%U A025936 293463446955454,3521586773279037,42259168371397574,507110656046025629
%N A025936 Expansion of 1/((1-2x)(1-3x)(1-5x)(1-12x)).
%K A025936 nonn
%O A025936 0,2
%A A025936 njas
%I A048376
%S A048376 1,22,333,4444,55555,666666,7777777,88888888,999999999,1,11,122,1333,
%T A048376 14444,155555,1666666,17777777,188888888,1999999999,22,221,2222,22333,
%U A048376 224444,2255555,22666666,227777777,2288888888,22999999999,333,3331
%N A048376 Replace each 1 in decimal expansion of n with 1 1's, each 2 with 2 2's, etc. (0 vanishes).
%e A048376 12 -> 122, 123->122333.
%Y A048376 Cf. A048377.
%K A048376 nonn,easy,base,nice
%O A048376 1,2
%A A048376 Patrick De Geest (pdg@worldofnumbers.com), Mar 1999.
%I A053422
%S A053422 0,1,22,333,4444,55555,666666,7777777,88888888,999999999,1111111110,
%T A053422 12222222221,133333333332,1444444444443,15555555555554,166666666666665,
%U A053422 1777777777777776,18888888888888887,199999999999999998
%N A053422 n times (n 1's): n*(10^n - 1)/9.
%F A053422 a(n) = n*A002275(n) = a(n-1)*10n/(n-1)+n
%Y A053422 Cf. A000461, A048376.
%K A053422 base,easy,nonn
%O A053422 0,3
%A A053422 Henry Bottomley (se16@btinternet.com), Mar 07 2000
%I A000461
%S A000461 1,22,333,4444,55555,666666,7777777,88888888,999999999,
%T A000461 10101010101010101010,1111111111111111111111,
%U A000461 121212121212121212121212,13131313131313131313131313
%N A000461 Concatenate n n times.
%D A000461 F. Smarandache, "Properties of the numbers", Univ. of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.
%H A000461 M. L. Perez et al., eds., Smarandache Notions Journal
%H A000461 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A000461 a(n)=n*(10^(n*L(n))-1)/(10^L(n)-1) where L(n)=A004216(n)+1=floor[log10(10n)] - Henry Bottomley (se16@btinternet.com), Jun 01 2000
%Y A000461 Cf. A048376, A053422.
%K A000461 nonn,base,easy
%O A000461 1,2
%A A000461 John Radu (Suttones@aol.com)
%I A048795
%S A048795 22,333,55555,7777777,1111111111111111111111,
%T A048795 13131313131313131313131313,1717171717171717171717171717171717,
%U A048795 19191919191919191919191919191919191919
%N A048795 Concatenate p p times, where p runs through the primes.
%H A048795 M. L. Perez et al., eds., Smarandache Notions Journal
%Y A048795 Cf. A000461.
%K A048795 nonn,base,easy
%O A048795 1,1
%A A048795 Patrick De Geest (pdg@worldofnumbers.com), Jul 1999.
%I A021284
%S A021284 1,22,335,4400,53481,620202,6970675,76624900,828512861,
%T A021284 8845504382,93498427815,980374738200,10212261530641,105799242660562,
%U A021284 1091082072825755,11208627544304300,114766536787594821
%N A021284 Expansion of 1/((1-x)(1-2x)(1-9x)(1-10x)).
%K A021284 nonn
%O A021284 0,2
%A A021284 njas
%I A019623
%S A019623 1,22,335,4400,53661,628122,7178395,80862100,902846921,
%T A019623 10025125022,110934086055,1224883116600,13505988249781,
%U A019623 148791855626722,1638292574483315,18032294531183900,198432777621062241
%N A019623 Expansion of 1/((1-4x)(1-7x)(1-11x)).
%K A019623 nonn
%O A019623 0,2
%A A019623 njas
%I A021794
%S A021794 1,22,335,4460,56061,686802,8317435,100210120,1204613321,
%T A021794 14466168782,173649468135,2084076423780,25010353485781,
%U A021794 300131513309962,3601614875036435,43219563508677440,518635692871953441
%N A021794 Expansion of 1/((1-x)(1-4x)(1-5x)(1-12x)).
%K A021794 nonn
%O A021794 0,2
%A A021794 njas
%I A018090
%S A018090 1,22,337,4450,54301,631462,7111417,78287530,847442101,
%T A018090 9055541902,95785566097,1004927161810,10472915657101,108541954516342,
%U A018090 1119734731454377,11506184005511290,117841370316867301
%N A018090 Expansion of 1/((1-3x)(1-9x)(1-10x)).
%K A018090 nonn
%O A018090 0,2
%A A018090 njas
%I A021274
%S A021274 1,22,337,4482,55533,660774,7667929,87542794,988535845,
%T A021274 11078416206,123498755601,1371575734386,15192048468637,
%U A021274 167950256294518,1854154604388553,20449314929530458,225371378475017109
%N A021274 Expansion of 1/((1-x)(1-2x)(1-8x)(1-11x)).
%K A021274 nonn
%O A021274 0,2
%A A021274 njas
%I A021534
%S A021534 1,22,337,4522,57253,705334,8574889,103567234,1246828045,
%T A021534 14986093486,179978152081,2160608272186,25932522746677,
%U A021534 311221616234278,3734847461630713,44819297962008178,537838346143305949
%N A021534 Expansion of 1/((1-x)(1-3x)(1-6x)(1-12x)).
%K A021534 nonn
%O A021534 0,2
%A A021534 njas
%I A018070
%S A018070 1,22,339,4532,56357,672210,7813303,89300464,1009144713,
%T A018070 11315328398,126186563867,1401795991596,15529706751469,
%U A018070 171706382611786,1895807080277631,20910172869786728,230462261504563025
%N A018070 Expansion of 1/((1-3x)(1-8x)(1-11x)).
%K A018070 nonn
%O A018070 0,2
%A A018070 njas
%I A019490
%S A019490 1,22,340,4600,58576,724192,8822080,106672000,1284971776,
%T A019490 15449370112,185571742720,2227940915200,26741787774976,
%U A019490 320940501164032,3851520569589760,46219655242547200,554644317650354176
%N A019490 Expansion of 1/((1-4x)(1-6x)(1-12x)).
%K A019490 nonn
%O A019490 0,2
%A A019490 njas
%I A021254
%S A021254 1,22,341,4646,59661,743358,9112405,110693878,1337742173,
%T A021254 16118816558,193887174117,2329875721446,27981116089837,
%U A021254 335931645121822,4032287505801077,48395204420052950,580796733493846653
%N A021254 Expansion of 1/((1-x)(1-2x)(1-7x)(1-12x)).
%K A021254 nonn
%O A021254 0,2
%A A021254 njas
%I A018055
%S A018055 1,22,343,4696,60493,755146,9267091,112644652,1361819305,
%T A018055 16412435710,197443515919,2372782379968,28497610413637,
%U A018055 342140879536114,4106877441223627,49290837516564244,591548207794982689
%N A018055 Expansion of 1/((1-3x)(1-7x)(1-12x)).
%K A018055 nonn
%O A018055 0,2
%A A018055 njas
%I A016322
%S A016322 1,22,345,4730,60461,740982,8834065,103324210,1191912021,
%T A016322 13609144142,154183593785,1736366607690,19463156373181,
%U A016322 217362833310502,2420404185281505,26889163207923170,298163249815659941
%N A016322 Expansion of 1/((1-2x)(1-9x)(1-11x)).
%K A016322 nonn
%O A016322 0,2
%A A016322 njas
%I A016320
%S A016320 1,22,348,4856,63728,808416,10050496,123402112,1503194880,
%T A016320 18217295360,220039199744,2651923642368,31914709676032,
%U A016320 383709523861504,4610378348347392,55371452676276224,664832732084240384
%N A016320 Expansion of 1/((1-2x)(1-8x)(1-12x)).
%K A016320 nonn
%O A016320 0,2
%A A016320 njas
%I A036738
%S A036738 1,1,1,22,351,5027,74362,1174272,19286295,323373490,5512884809,
%T A036738 95418481246,1672641384984,29627580485085,529439149790790,9533475917817236,
%U A036738 172815097505687679,3151049783203470555,57753787348458752554,1063443233267664064332
%N A036738 G.f. satisfies A(x) = 1 + x*cycle_index(G,A(x)) where G = cyclic group of order 43 generated by (1,2,...,43).
%Y A036738 Cf. A036717-A036726.
%K A036738 nonn
%O A036738 0,4
%A A036738 njas
%I A054940
%S A054940 0,1,1,22,352,13412,932768,125776816,33148125376,17248244478848,
%T A054940 17820828772662272,36677298103403269504,150636101324831804975104,
%U A054940 1235824405515221870758956032,20263840468865240117261288526848
%N A054940 Connected labeled graphs with n nodes and an odd number of edges.
%H A054940 V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.
%F A054940 a(n)=(A001187(n)+(-1)^n*A000142(n-1))/2.
%Y A054940 Cf. A054939.
%K A054940 nonn,easy
%O A054940 1,4
%A A054940 njas, May 24 2000
%E A054940 More terms from Vladeta Jovovic (vladeta@Eunet.yu), Jul 17 2000
%I A016265
%S A016265 1,22,353,4994,66045,837606,10324777,124683658,1482631349,
%T A016265 17420055950,202731726561,2341160103282,26863872247213,
%U A016265 306613705830454,3483861875246105,39433591738818266,444880620238112037
%N A016265 Expansion of 1/((1-x)(1-10x)(1-11x)).
%K A016265 nonn
%O A016265 0,2
%A A016265 njas
%I A016263
%S A016263 1,22,355,5080,68341,886522,11236135,140214460,1731001081,
%T A016263 21207861022,258416964715,3136307268640,37953420452221,
%U A016263 458300644483522,5525344125314095,66535757027375620,800513732040965761
%N A016263 Expansion of 1/((1-x)(1-9x)(1-12x)).
%K A016263 nonn
%O A016263 0,2
%A A016263 njas
%I A001718 M5127 N2222
%S A001718 1,22,355,5265,77224,1155420,17893196,288843260,4876196776,
%T A001718 86194186584,1595481972864,30908820004608,626110382381184
%N A001718 Generalized Stirling numbers.
%D A001718 Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%K A001718 nonn
%O A001718 0,2
%A A001718 njas
%I A016196
%S A016196 1,22,364,5368,74416,992992,12915904,164990848,2079890176,
%T A016196 25958682112,321504185344,3958050224128,48496602689536,
%U A016196 591959232274432,7203510787293184,87442129447518208,1059305553370218496
%N A016196 Expansion of 1/((1-10x)(1-12x)).
%K A016196 nonn
%O A016196 0,2
%A A016196 njas
%I A049663
%S A049663 1,22,399,7164,128557,2306866,41395035,742803768,13329072793,
%T A049663 239180506510,4291920044391,77015380292532,1381984925221189,
%U A049663 24798713273688874,444994854001178547,7985108658747524976
%N A049663 a(n)=(F(6n+5)-1)/4, where F=A000045 (the Fibonacci sequence).
%K A049663 nonn
%O A049663 0,2
%A A049663 Clark Kimberling, ck6@cedar.evansville.edu
%E A049663 More terms from James A. Sellers (sellersj@math.psu.edu), Jan 20 2000
%I A014904
%S A014904 1,22,443,8864,177285,3545706,70914127,1418282548,28365650969,
%T A014904 567313019390,11346260387811,226925207756232,4538504155124653,
%U A014904 90770083102493074,1815401662049861495,36308033240997229916
%N A014904 a(1)=1, a(n)=20*a(n-1)+n.
%K A014904 nonn
%O A014904 1,2
%A A014904 njas, Olivier Gerard (ogerard@ext.jussieu.fr)
%I A036905
%S A036905 22,444,7655,98370,1107795,12983306
%N A036905 Scan decimal expansion of ln(2) until all n-digit strings have been seen; a(n) is number of digits that must be scanned.
%Y A036905 Cf. A032510, A036900-A036906.
%K A036905 nonn
%O A036905 1,1
%A A036905 Michael Kleber (kleber@math.mit.edu)
%I A009966
%S A009966 1,22,484,10648,234256,5153632,113379904,2494357888,54875873536,
%T A009966 1207269217792,26559922791424,584318301411328,12855002631049216,
%U A009966 282810057883082752,6221821273427820544,136880068015412051968
%N A009966 Powers of 22.
%K A009966 nonn
%O A009966 0,2
%A A009966 njas
%I A041221
%S A041221 1,22,485,10692,235709,5196290,114554089,2525386248,55673051545,
%T A041221 1227332520238,27056988496781,596481079449420,13149640736384021,
%U A041221 289888577279897882,6390698340894137425,140885252076950921232
%N A041221 Denominators of continued fraction convergents to sqrt(122).
%Y A041221 Cf. A041220.
%K A041221 nonn,cofr,easy
%O A041221 0,2
%A A041221 njas
%I A041926
%S A041926 22,485,21362,470449,20721118,456335045,20099463098,442644523201,
%T A041926 19496458483942,429364731169925,18911544629960642,416483346590304049,
%U A041926 18344178794603338798,403988416827863757605,17793834519220608673418
%N A041926 Numerators of continued fraction convergents to sqrt(486).
%Y A041926 Cf. A041927.
%K A041926 nonn,cofr,easy
%O A041926 0,1
%A A041926 njas
%I A025760
%S A025760 1,1,22,680,24074,917414,36618492,1508943612,63643109727,
%T A025760 2732349490669,118957846271104,5237911268468572,232794783971436296,
%U A025760 10427673857731312064,470213556090357498728,21325335129901497816528
%N A025760 7th order Vatalan numbers (generalization of Catalan numbers).
%F A025760 G.f.: ${7\over {6+\root 7 \of {1-49x}}}$
%K A025760 nonn
%O A025760 0,3
%A A025760 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A027409
%S A027409 1,22,715,28974,1380093,74841558,4525209711,300561720030,
%T A027409 21686284145097,1685085369468534,140028734022344739,
%U A027409 12373253024424889518,1157017818664396423845,114029077860692830245366
%N A027409 Labeled servers of dimension 22.
%D A027409 R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
%F A027409 E.g.f.: Exp[ Sum[ ((1+x)^i-1)/i,{i, 1, 22} ] ]
%K A027409 nonn
%O A027409 0,2
%A A027409 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A041927
%S A041927 1,22,969,21340,939929,20699778,911730161,20078763320,884377316241,
%T A041927 19476379720622,857845085023609,18892068250240020,832108848095584489,
%U A041927 18325286726353098778,807144724807631930721,17775509232494255574640
%N A041927 Denominators of continued fraction convergents to sqrt(486).
%Y A041927 Cf. A041926.
%K A041927 nonn,cofr,easy
%O A041927 0,2
%A A041927 njas
%I A041924
%S A041924 22,969,42658,1877921,82671182,3639409929,160216708058,
%T A041924 7053174564481,310499897545222,13669048666554249,601748641225932178,
%U A041924 26490609262607570081,1166188556195959015742,51338787081884804262729
%N A041924 Numerators of continued fraction convergents to sqrt(485).
%Y A041924 Cf. A041925.
%K A041924 nonn,cofr,easy
%O A041924 0,1
%A A041924 njas
%I A004633
%S A004633 1,22,1000,2101,11122,22000,110201,200222,1000000,1101001,
%T A004633 1211022,2101000,10000101,10202122,11122000,12121201,20201222,
%U A004633 22000000,100102001,101222022,110201000,112121101,121200122
%N A004633 Cubes written in base 3.
%K A004633 nonn
%O A004633 0,2
%A A004633 njas
%I A038695
%S A038695 22,1111,101909,10097063,1000730021,100003300009,10000049000057,
%T A038695 1000001970000133,100000007700000049,10000000089000000133,
%U A038695 1000000001930000000057,100000000006900000000117
%N A038695 Smallest n-digit prime * smallest (n+1)-digit prime.
%K A038695 nonn,base
%O A038695 1,1
%A A038695 Jeff Burch (gburch@erols.com)
%I A055475
%S A055475 1,22,1210,33220,2130100,120122200,3310021000,212021122000,
%T A055475 11331132010000,323212230220000,21110002332100000,1131020131232200000,
%U A055475 32203110221101000000,2101201032130222000000,112233030100132210000000
%N A055475 Powers of ten written in base 4.
%Y A055475 Cf. A000468, A011557.
%K A055475 base,easy,nonn
%O A055475 0,2
%A A055475 Henry Bottomley (se16@btinternet.com), Jun 27 2000
%E A055475 More terms from James A. Sellers (sellersj@math.psu.edu), Jul 05 2000
%I A033526
%S A033526 22,1511,90040,5493583,334056618,20324827981,1236501116120,
%T A033526 75226160041933,4576591071807054,278429681683117411,
%U A033526 16939044773645481920,1030533959174319758227,62695402974582513118434
%N A033526 The number of matchings in graph P_{2} X P_{3} X P_{n}
%D A033526 Per Hakan Lundow, "Computation of matching polynomials and the number of 1-factors in polygraphs", Research reports, No 12, 1996, Department of Mathematics, Umea University.
%H A033526 P.-H. Lundow, Enumeration of matchings in polygraphs, 1998.
%K A033526 nonn
%O A033526 0,1
%A A033526 P.H. Lundow (per-hakan.lundow@math.umu.se)
%I A046445
%S A046445 1,22,2222,2241998,22435673986,2243634705621958,
%T A046445 22436389685278986817202,2243639125582626478673107720414,
%U A046445 2243639141288100357751493071125754042898
%N A046445 Smallest composite with n prime factors that are distinct in length.
%Y A046445 Cf. A003617, A033873. Initial terms of A046442, A046443, A046444.
%K A046445 nonn
%O A046445 0,2
%A A046445 Patrick De Geest (pdg@worldofnumbers.com), Jul 1998.
%I A013727
%S A013727 22,10648,5153632,2494357888,1207269217792,584318301411328,
%T A013727 282810057883082752,136880068015412051968,66249952919459433152512,
%U A013727 32064977213018365645815808,15519448971100888972574851072
%N A013727 22^(2n+1).
%K A013727 nonn,easy
%O A013727 0,1
%A A013727 njas
%I A028692
%S A028692 1,22,11616,141320256,39547060439040,254538406080331591680,
%T A028692 37680818974206486508802211840,128296611269497862923425473853914480640,
%U A028692 10047034036599529256387830050150921763777884979200
%N A028692 23-factorial numbers.
%H A028692 Index entries for sequences related to factorial numbers
%t A028692 FoldList[ #1 (23^#2-1)&,1,Range[ 20 ] ]
%K A028692 nonn
%O A028692 0,2
%A A028692 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A060619
%S A060619 0,1,22,52224
%N A060619 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=9 and D varies.
%D A060619 A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
%D A060619 N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octogonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
%D A060619 Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.
%H A060619 M. Latapy, Tilings of Zonotopes
%e A060619 For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 1.
%Y A060619 Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.
%K A060619 nonn
%O A060619 9,3
%A A060619 Matthieu Latapy (latapy@liafa.jussieu.fr), Apr 13 2001
%I A013770
%S A013770 22,234256,2494357888,26559922791424,282810057883082752,
%T A013770 3011361496339065143296,32064977213018365645815808,341427877364219557396646723584,
%U A013770 3635524038174209847159494312722432,38711059958478986452554295441868455936
%N A013770 22^(3n+1).
%K A013770 nonn
%O A013770 0,1
%A A013770 njas
%I A013816
%S A013816 22,5153632,1207269217792,282810057883082752,66249952919459433152512,
%T A013816 15519448971100888972574851072,3635524038174209847159494312722432,
%U A013816 851643319086537701956194499721106030592,199502557355935975909450298726667414302359552
%N A013816 22^(4n+1).
%K A013816 nonn
%O A013816 0,1
%A A013816 njas
%I A047841
%S A047841 22,10213223,10311233,10313314,10313315,10313316,10313317,10313318,
%T A047841 10313319,21322314,21322315,21322316,21322317,21322318,21322319,
%U A047841 31123314,31123315,31123316,31123317,31123318,31123319
%N A047841 Fixed under operator T (A047842): "Say what you see".
%C A047841 a(n) is finite, since T(x) < x for every x with at least 22 digits. Last term is a(109)=101112213141516171819.
%e A047841 10313314 contains 1 0's, 3 1's, 3 3's and 1 4's, hence T(10313314) = 10313314 is in the sequence
%Y A047841 Cf. A005151, which is the sequence 1, T(1), T(T(1)), .. ending in the fixed-point 21322314.
%K A047841 nonn,fini,base,nice,eigen
%O A047841 1,1
%A A047841 Ulrich Schimke (ulrich_schimke@idg.de)
%I A013902
%S A013902 22,113379904,584318301411328,3011361496339065143296,15519448971100888972574851072,
%T A013902 79981528839832616637508874879893504,412195366437884247746798137865015318806528,
%U A013902 2124303230726006271483826780841554627491524509696,10947877107572929152919737180202022857988400441953615872
%N A013902 22^(5n+1).
%K A013902 nonn
%O A013902 0,1
%A A013902 njas
%I A056667
%S A056667 22,47093135946,886151997189981695113820646176823296,
%T A056667 3134651602192766687916630765925989003578700185329945310286348103545655409509684765625,
%U A056667 929105607753267830402847266093582872945009794334454648424476250883505063045211468786079180624616064970104635594329005644118117051244756456544404209339748566921547776
%N A056667 Equivalence classes of n-valued Post functions of 3 variables under action of semi-direct product of symmetric group S_3 and complementing group C(3,n).
%D A056667 M. H. Harrison and R. G. High, On the cycle index of a product of permutation groups, J. Combin. Theory, 4 (1968), p. 296.
%F A056667 I. Strazdins, On the number of types of l-adic functions (in Russian), Proc. Riga Polytech. Inst. 10, No. 1 (1963), 167-186.
%Y A056667 Cf. A000616, A001321-A001324.
%K A056667 huge,nonn
%O A056667 2,1
%A A056667 Vladeta Jovovic (vladeta@Eunet.yu), Aug 09 2000
%I A040534
%S A040534 23,1,1,1,1,1,4,1,1,1,1,1,46,1,1,1,1,1,4,1,1,1,1,1,46,1,1,1,1,1,
%T A040534 4,1,1,1,1,1,46,1,1,1,1,1,4,1,1,1,1,1,46,1,1,1,1,1,4,1,1,1,1,1,
%U A040534 46,1,1,1,1,1,4,1,1,1,1,1,46,1,1,1,1,1,4,1,1,1,1,1,46,1,1,1,1,1
%N A040534 Continued fraction for sqrt(558).
%H A040534 Index entries for continued fractions for constants
%p A040534 with(numtheory): Digits:=300: convert(evalf(sqrt(558)),confrac);
%K A040534 nonn,cofr,easy
%O A040534 0,1
%A A040534 njas
%I A040533
%S A040533 23,1,1,1,1,46,1,1,1,1,46,1,1,1,1,46,1,1,1,1,46,1,1,1,1,46,1,1,
%T A040533 1,1,46,1,1,1,1,46,1,1,1,1,46,1,1,1,1,46,1,1,1,1,46,1,1,1,1,46,
%U A040533 1,1,1,1,46,1,1,1,1,46,1,1,1,1,46,1,1,1,1,46,1,1,1,1,46,1,1,1,1
%N A040533 Continued fraction for sqrt(557).
%H A040533 Index entries for continued fractions for constants
%p A040533 with(numtheory): Digits:=300: convert(evalf(sqrt(557)),confrac);
%K A040533 nonn,cofr,easy
%O A040533 0,1
%A A040533 njas
%I A040535
%S A040535 23,1,1,1,4,15,1,1,4,1,2,1,4,1,1,15,4,1,1,1,46,1,1,1,4,15,1,1,4,
%T A040535 1,2,1,4,1,1,15,4,1,1,1,46,1,1,1,4,15,1,1,4,1,2,1,4,1,1,15,4,1,
%U A040535 1,1,46,1,1,1,4,15,1,1,4,1,2,1,4,1,1,15,4,1,1,1,46,1,1,1,4,15,1
%N A040535 Continued fraction for sqrt(559).
%H A040535 Index entries for continued fractions for constants
%p A040535 with(numtheory): Digits:=300: convert(evalf(sqrt(559)),confrac);
%K A040535 nonn,cofr,easy
%O A040535 0,1
%A A040535 njas
%I A040536
%S A040536 23,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,
%T A040536 46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,
%U A040536 46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1,46,1,1,1
%N A040536 Continued fraction for sqrt(560).
%H A040536 Index entries for continued fractions for constants
%p A040536 with(numtheory): Digits:=300: convert(evalf(sqrt(560)),confrac);
%K A040536 nonn,cofr,easy
%O A040536 0,1
%A A040536 njas
%I A040532
%S A040532 23,1,1,2,1,1,1,3,3,2,1,5,5,15,1,1,8,1,10,1,8,1,1,15,5,5,1,2,3,
%T A040532 3,1,1,1,2,1,1,46,1,1,2,1,1,1,3,3,2,1,5,5,15,1,1,8,1,10,1,8,1,1,
%U A040532 15,5,5,1,2,3,3,1,1,1,2,1,1,46,1,1,2,1,1,1,3,3,2,1,5,5,15,1,1,8
%N A040532 Continued fraction for sqrt(556).
%H A040532 Index entries for continued fractions for constants
%p A040532 with(numtheory): Digits:=300: convert(evalf(sqrt(556)),confrac);
%K A040532 nonn,cofr,easy
%O A040532 0,1
%A A040532 njas
%I A040531
%S A040531 23,1,1,3,1,3,1,1,46,1,1,3,1,3,1,1,46,1,1,3,1,3,1,1,46,1,1,3,1,
%T A040531 3,1,1,46,1,1,3,1,3,1,1,46,1,1,3,1,3,1,1,46,1,1,3,1,3,1,1,46,1,
%U A040531 1,3,1,3,1,1,46,1,1,3,1,3,1,1,46,1,1,3,1,3,1,1,46,1,1,3,1,3,1,1
%N A040531 Continued fraction for sqrt(555).
%H A040531 Index entries for continued fractions for constants
%p A040531 with(numtheory): Digits:=300: convert(evalf(sqrt(555)),confrac);
%K A040531 nonn,cofr,easy
%O A040531 0,1
%A A040531 njas
%I A040530
%S A040530 23,1,1,6,4,1,1,4,6,1,1,46,1,1,6,4,1,1,4,6,1,1,46,1,1,6,4,1,1,4,
%T A040530 6,1,1,46,1,1,6,4,1,1,4,6,1,1,46,1,1,6,4,1,1,4,6,1,1,46,1,1,6,4,
%U A040530 1,1,4,6,1,1,46,1,1,6,4,1,1,4,6,1,1,46,1,1,6,4,1,1,4,6,1,1,46,1
%N A040530 Continued fraction for sqrt(554).
%H A040530 Index entries for continued fractions for constants
%p A040530 with(numtheory): Digits:=300: convert(evalf(sqrt(554)),confrac);
%K A040530 nonn,cofr,easy
%O A040530 0,1
%A A040530 njas
%I A040529
%S A040529 23,1,1,15,5,1,4,2,1,1,3,1,2,6,2,1,3,1,1,2,4,1,5,15,1,1,46,1,1,
%T A040529 15,5,1,4,2,1,1,3,1,2,6,2,1,3,1,1,2,4,1,5,15,1,1,46,1,1,15,5,1,
%U A040529 4,2,1,1,3,1,2,6,2,1,3,1,1,2,4,1,5,15,1,1,46,1,1,15,5,1,4,2,1,1
%N A040529 Continued fraction for sqrt(553).
%H A040529 Index entries for continued fractions for constants
%p A040529 with(numtheory): Digits:=300: convert(evalf(sqrt(553)),confrac);
%K A040529 nonn,cofr,easy
%O A040529 0,1
%A A040529 njas
%I A022186
%S A022186 1,1,1,1,23,1,1,507,507,1,1,11155,245895,11155,1,1,245411,
%T A022186 119024335,119024335,245411,1,1,5399043,57608023551,1267490143415,
%U A022186 57608023551,5399043,1,1,118778947,27882288797727,13496292655106471
%N A022186 Triangle of Gaussian binomial coefficients [ n,k ] for q = 22.
%D A022186 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%K A022186 nonn,tabl
%O A022186 0,5
%A A022186 njas
%I A015151
%S A015151 1,1,1,1,23,1,1,553,553,1,1,13271,319081,13271,1,1,318505,183777385,
%T A015151 183777385,318505,1,1,7644119,105856092265,2540354792855,105856092265,
%U A015151 7644119,1,1,183458857,60973101500521,35117970512519785
%V A015151 1,1,1,1,-23,1,1,553,553,1,1,-13271,319081,-13271,1,1,318505,183777385,
%W A015151 183777385,318505,1,1,-7644119,105856092265,-2540354792855,105856092265,
%X A015151 -7644119,1,1,183458857,60973101500521,35117970512519785
%N A015151 Triangle of (Gaussian) q-binomial coefficients for q=-24.
%K A015151 sign,done,tabl,easy
%O A015151 0,5
%A A015151 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A040539
%S A040539 23,1,2,1,2,23,2,1,2,1,46,1,2,1,2,23,2,1,2,1,46,1,2,1,2,23,2,1,
%T A040539 2,1,46,1,2,1,2,23,2,1,2,1,46,1,2,1,2,23,2,1,2,1,46,1,2,1,2,23,
%U A040539 2,1,2,1,46,1,2,1,2,23,2,1,2,1,46,1,2,1,2,23,2,1,2,1,46,1,2,1,2
%N A040539 Continued fraction for sqrt(563).
%H A040539 Index entries for continued fractions for constants
%p A040539 with(numtheory): Digits:=300: convert(evalf(sqrt(563)),confrac);
%K A040539 nonn,cofr,easy
%O A040539 0,1
%A A040539 njas
%I A040540
%S A040540 23,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,
%T A040540 46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,
%U A040540 46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1,46,1,2,1
%N A040540 Continued fraction for sqrt(564).
%H A040540 Index entries for continued fractions for constants
%p A040540 with(numtheory): Digits:=300: convert(evalf(sqrt(564)),confrac);
%K A040540 nonn,cofr,easy
%O A040540 0,1
%A A040540 njas
%I A040538
%S A040538 23,1,2,2,2,4,1,5,1,22,1,5,1,4,2,2,2,1,46,1,2,2,2,4,1,5,1,22,1,
%T A040538 5,1,4,2,2,2,1,46,1,2,2,2,4,1,5,1,22,1,5,1,4,2,2,2,1,46,1,2,2,2,
%U A040538 4,1,5,1,22,1,5,1,4,2,2,2,1,46,1,2,2,2,4,1,5,1,22,1,5,1,4,2,2,2
%N A040538 Continued fraction for sqrt(562).
%H A040538 Index entries for continued fractions for constants
%p A040538 with(numtheory): Digits:=300: convert(evalf(sqrt(562)),confrac);
%K A040538 nonn,cofr,easy
%O A040538 0,1
%A A040538 njas
%I A040537
%S A040537 23,1,2,5,1,1,2,2,2,1,1,5,2,1,46,1,2,5,1,1,2,2,2,1,1,5,2,1,46,1,
%T A040537 2,5,1,1,2,2,2,1,1,5,2,1,46,1,2,5,1,1,2,2,2,1,1,5,2,1,46,1,2,5,
%U A040537 1,1,2,2,2,1,1,5,2,1,46,1,2,5,1,1,2,2,2,1,1,5,2,1,46,1,2,5,1,1
%N A040537 Continued fraction for sqrt(561).
%H A040537 Index entries for continued fractions for constants
%p A040537 with(numtheory): Digits:=300: convert(evalf(sqrt(561)),confrac);
%K A040537 nonn,cofr,easy
%O A040537 0,1
%A A040537 njas
%I A040542
%S A040542 23,1,3,1,3,1,1,8,1,22,1,8,1,1,3,1,3,1,46,1,3,1,3,1,1,8,1,22,1,
%T A040542 8,1,1,3,1,3,1,46,1,3,1,3,1,1,8,1,22,1,8,1,1,3,1,3,1,46,1,3,1,3,
%U A040542 1,1,8,1,22,1,8,1,1,3,1,3,1,46,1,3,1,3,1,1,8,1,22,1,8,1,1,3,1,3
%N A040542 Continued fraction for sqrt(566).
%H A040542 Index entries for continued fractions for constants
%p A040542 with(numtheory): Digits:=300: convert(evalf(sqrt(566)),confrac);
%K A040542 nonn,cofr,easy
%O A040542 0,1
%A A040542 njas
%I A040541
%S A040541 23,1,3,2,1,11,5,5,11,1,2,3,1,46,1,3,2,1,11,5,5,11,1,2,3,1,46,1,
%T A040541 3,2,1,11,5,5,11,1,2,3,1,46,1,3,2,1,11,5,5,11,1,2,3,1,46,1,3,2,
%U A040541 1,11,5,5,11,1,2,3,1,46,1,3,2,1,11,5,5,11,1,2,3,1,46,1,3,2,1,11
%N A040541 Continued fraction for sqrt(565).
%H A040541 Index entries for continued fractions for constants
%p A040541 with(numtheory): Digits:=300: convert(evalf(sqrt(565)),confrac);
%K A040541 nonn,cofr,easy
%O A040541 0,1
%A A040541 njas
%I A040544
%S A040544 23,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,
%T A040544 46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,
%U A040544 46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1,46,1,4,1
%N A040544 Continued fraction for sqrt(568).
%H A040544 Index entries for continued fractions for constants
%p A040544 with(numtheory): Digits:=300: convert(evalf(sqrt(568)),confrac);
%K A040544 nonn,cofr,easy
%O A040544 0,1
%A A040544 njas
%I A040543
%S A040543 23,1,4,3,4,1,46,1,4,3,4,1,46,1,4,3,4,1,46,1,4,3,4,1,46,1,4,3,4,
%T A040543 1,46,1,4,3,4,1,46,1,4,3,4,1,46,1,4,3,4,1,46,1,4,3,4,1,46,1,4,3,
%U A040543 4,1,46,1,4,3,4,1,46,1,4,3,4,1,46,1,4,3,4,1,46,1,4,3,4,1,46,1,4
%N A040543 Continued fraction for sqrt(567).
%H A040543 Index entries for continued fractions for constants
%p A040543 with(numtheory): Digits:=300: convert(evalf(sqrt(567)),confrac);
%K A040543 nonn,cofr,easy
%O A040543 0,1
%A A040543 njas
%I A040545
%S A040545 23,1,5,1,5,9,2,1,2,3,3,2,1,2,9,5,1,5,1,46,1,5,1,5,9,2,1,2,3,3,
%T A040545 2,1,2,9,5,1,5,1,46,1,5,1,5,9,2,1,2,3,3,2,1,2,9,5,1,5,1,46,1,5,
%U A040545 1,5,9,2,1,2,3,3,2,1,2,9,5,1,5,1,46,1,5,1,5,9,2,1,2,3,3,2,1,2,9
%N A040545 Continued fraction for sqrt(569).
%H A040545 Index entries for continued fractions for constants
%p A040545 with(numtheory): Digits:=300: convert(evalf(sqrt(569)),confrac);
%K A040545 nonn,cofr,easy
%O A040545 0,1
%A A040545 njas
%I A040546
%S A040546 23,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,
%T A040546 46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,
%U A040546 46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1,46,1,6,1
%N A040546 Continued fraction for sqrt(570).
%H A040546 Index entries for continued fractions for constants
%p A040546 with(numtheory): Digits:=300: convert(evalf(sqrt(570)),confrac);
%K A040546 nonn,cofr,easy
%O A040546 0,1
%A A040546 njas
%I A040547
%S A040547 23,1,8,1,1,2,1,1,1,15,3,2,1,6,7,1,4,2,3,4,2,23,2,4,3,2,4,1,7,6,
%T A040547 1,2,3,15,1,1,1,2,1,1,8,1,46,1,8,1,1,2,1,1,1,15,3,2,1,6,7,1,4,2,
%U A040547 3,4,2,23,2,4,3,2,4,1,7,6,1,2,3,15,1,1,1,2,1,1,8,1,46,1,8,1,1,2
%N A040547 Continued fraction for sqrt(571).
%H A040547 Index entries for continued fractions for constants
%p A040547 with(numtheory): Digits:=300: convert(evalf(sqrt(571)),confrac);
%K A040547 nonn,cofr,easy
%O A040547 0,1
%A A040547 njas
%I A040548
%S A040548 23,1,10,1,46,1,10,1,46,1,10,1,46,1,10,1,46,1,10,1,46,1,10,1,46,
%T A040548 1,10,1,46,1,10,1,46,1,10,1,46,1,10,1,46,1,10,1,46,1,10,1,46,1,
%U A040548 10,1,46,1,10,1,46,1,10,1,46,1,10,1,46,1,10,1,46,1,10,1,46,1,10
%N A040548 Continued fraction for sqrt(572).
%H A040548 Index entries for continued fractions for constants
%p A040548 with(numtheory): Digits:=300: convert(evalf(sqrt(572)),confrac);
%K A040548 nonn,cofr,easy
%O A040548 0,1
%A A040548 njas
%I A040549
%S A040549 23,1,14,1,46,1,14,1,46,1,14,1,46,1,14,1,46,1,14,1,46,1,14,1,46,
%T A040549 1,14,1,46,1,14,1,46,1,14,1,46,1,14,1,46,1,14,1,46,1,14,1,46,1,
%U A040549 14,1,46,1,14,1,46,1,14,1,46,1,14,1,46,1,14,1,46,1,14,1,46,1,14
%N A040549 Continued fraction for sqrt(573).
%H A040549 Index entries for continued fractions for constants
%p A040549 with(numtheory): Digits:=300: convert(evalf(sqrt(573)),confrac);
%K A040549 nonn,cofr,easy
%O A040549 0,1
%A A040549 njas
%I A040550
%S A040550 23,1,22,1,46,1,22,1,46,1,22,1,46,1,22,1,46,1,22,1,46,1,22,1,46,
%T A040550 1,22,1,46,1,22,1,46,1,22,1,46,1,22,1,46,1,22,1,46,1,22,1,46,1,
%U A040550 22,1,46,1,22,1,46,1,22,1,46,1,22,1,46,1,22,1,46,1,22,1,46,1,22
%N A040550 Continued fraction for sqrt(574).
%H A040550 Index entries for continued fractions for constants
%p A040550 with(numtheory): Digits:=300: convert(evalf(sqrt(574)),confrac);
%K A040550 nonn,cofr,easy
%O A040550 0,1
%A A040550 njas
%I A040551
%S A040551 23,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,
%T A040551 1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,
%U A040551 46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46,1,46
%N A040551 Continued fraction for sqrt(575).
%H A040551 Index entries for continued fractions for constants
%p A040551 with(numtheory): Digits:=300: convert(evalf(sqrt(575)),confrac);
%K A040551 nonn,cofr,easy
%O A040551 0,1
%A A040551 njas
%I A040523
%S A040523 23,2,1,1,2,1,2,1,7,15,2,6,5,23,5,6,2,15,7,1,2,1,2,1,1,2,46,2,1,
%T A040523 1,2,1,2,1,7,15,2,6,5,23,5,6,2,15,7,1,2,1,2,1,1,2,46,2,1,1,2,1,
%U A040523 2,1,7,15,2,6,5,23,5,6,2,15,7,1,2,1,2,1,1,2,46,2,1,1,2,1,2,1,7
%N A040523 Continued fraction for sqrt(547).
%H A040523 Index entries for continued fractions for constants
%p A040523 with(numtheory): Digits:=300: convert(evalf(sqrt(547)),confrac);
%K A040523 nonn,cofr,easy
%O A040523 0,1
%A A040523 njas
%I A040522
%S A040522 23,2,1,2,1,2,46,2,1,2,1,2,46,2,1,2,1,2,46,2,1,2,1,2,46,2,1,2,1,
%T A040522 2,46,2,1,2,1,2,46,2,1,2,1,2,46,2,1,2,1,2,46,2,1,2,1,2,46,2,1,2,
%U A040522 1,2,46,2,1,2,1,2,46,2,1,2,1,2,46,2,1,2,1,2,46,2,1,2,1,2,46,2,1
%N A040522 Continued fraction for sqrt(546).
%H A040522 Index entries for continued fractions for constants
%p A040522 with(numtheory): Digits:=300: convert(evalf(sqrt(546)),confrac);
%K A040522 nonn,cofr,easy
%O A040522 0,1
%A A040522 njas
%I A040521
%S A040521 23,2,1,8,1,2,46,2,1,8,1,2,46,2,1,8,1,2,46,2,1,8,1,2,46,2,1,8,1,
%T A040521 2,46,2,1,8,1,2,46,2,1,8,1,2,46,2,1,8,1,2,46,2,1,8,1,2,46,2,1,8,
%U A040521 1,2,46,2,1,8,1,2,46,2,1,8,1,2,46,2,1,8,1,2,46,2,1,8,1,2,46,2,1
%N A040521 Continued fraction for sqrt(545).
%H A040521 Index entries for continued fractions for constants
%p A040521 with(numtheory): Digits:=300: convert(evalf(sqrt(545)),confrac);
%K A040521 nonn,cofr,easy
%O A040521 0,1
%A A040521 njas
%I A040524
%S A040524 23,2,2,3,1,5,1,10,1,5,1,3,2,2,46,2,2,3,1,5,1,10,1,5,1,3,2,2,46,
%T A040524 2,2,3,1,5,1,10,1,5,1,3,2,2,46,2,2,3,1,5,1,10,1,5,1,3,2,2,46,2,
%U A040524 2,3,1,5,1,10,1,5,1,3,2,2,46,2,2,3,1,5,1,10,1,5,1,3,2,2,46,2,2
%N A040524 Continued fraction for sqrt(548).
%H A040524 Index entries for continued fractions for constants
%p A040524 with(numtheory): Digits:=300: convert(evalf(sqrt(548)),confrac);
%K A040524 nonn,cofr,easy
%O A040524 0,1
%A A040524 njas
%I A040525
%S A040525 23,2,3,9,11,1,1,1,1,4,1,1,1,1,11,9,3,2,46,2,3,9,11,1,1,1,1,4,1,
%T A040525 1,1,1,11,9,3,2,46,2,3,9,11,1,1,1,1,4,1,1,1,1,11,9,3,2,46,2,3,9,
%U A040525 11,1,1,1,1,4,1,1,1,1,11,9,3,2,46,2,3,9,11,1,1,1,1,4,1,1,1,1,11
%N A040525 Continued fraction for sqrt(549).
%H A040525 Index entries for continued fractions for constants
%p A040525 with(numtheory): Digits:=300: convert(evalf(sqrt(549)),confrac);
%K A040525 nonn,cofr,easy
%O A040525 0,1
%A A040525 njas
%I A040526
%S A040526 23,2,4,1,2,1,1,7,4,7,1,1,2,1,4,2,46,2,4,1,2,1,1,7,4,7,1,1,2,1,
%T A040526 4,2,46,2,4,1,2,1,1,7,4,7,1,1,2,1,4,2,46,2,4,1,2,1,1,7,4,7,1,1,
%U A040526 2,1,4,2,46,2,4,1,2,1,1,7,4,7,1,1,2,1,4,2,46,2,4,1,2,1,1,7,4,7
%N A040526 Continued fraction for sqrt(550).
%H A040526 Index entries for continued fractions for constants
%p A040526 with(numtheory): Digits:=300: convert(evalf(sqrt(550)),confrac);
%K A040526 nonn,cofr,easy
%O A040526 0,1
%A A040526 njas
%I A040527
%S A040527 23,2,8,1,8,2,46,2,8,1,8,2,46,2,8,1,8,2,46,2,8,1,8,2,46,2,8,1,8,
%T A040527 2,46,2,8,1,8,2,46,2,8,1,8,2,46,2,8,1,8,2,46,2,8,1,8,2,46,2,8,1,
%U A040527 8,2,46,2,8,1,8,2,46,2,8,1,8,2,46,2,8,1,8,2,46,2,8,1,8,2,46,2,8
%N A040527 Continued fraction for sqrt(551).
%H A040527 Index entries for continued fractions for constants
%p A040527 with(numtheory): Digits:=300: convert(evalf(sqrt(551)),confrac);
%K A040527 nonn,cofr,easy
%O A040527 0,1
%A A040527 njas
%I A040528
%S A040528 23,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,
%T A040528 2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,
%U A040528 46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46,2,46
%N A040528 Continued fraction for sqrt(552).
%H A040528 Index entries for continued fractions for constants
%p A040528 with(numtheory): Digits:=300: convert(evalf(sqrt(552)),confrac);
%K A040528 nonn,cofr,easy
%O A040528 0,1
%A A040528 njas
%I A051313
%S A051313 23,2,47,3,13,84319,7109609443,463,23403050994721829453179,7,5,57367,
%T A051313 239,40237,10575444619218059847586376042094152838881224222904607376771,
%U A051313 31333,742759,9444637217
%N A051313 Euclid-Mullin sequence (A000945) with initial value a(1)=23 instead of a(1)=2.
%t A051313 a[ n_+1 ] :=First[ Flatten[ FactorInteger[ 1+Product[ a[ j ],{j,1,n} ] ] ] ]
%Y A051313 A000945, A000946, A005265, A005266.
%K A051313 nonn
%O A051313 1,1
%A A051313 Labos E. (labos@ana1.sote.hu)
%I A040518
%S A040518 23,3,1,1,3,1,1,1,22,1,1,1,3,1,1,3,46,3,1,1,3,1,1,1,22,1,1,1,3,
%T A040518 1,1,3,46,3,1,1,3,1,1,1,22,1,1,1,3,1,1,3,46,3,1,1,3,1,1,1,22,1,
%U A040518 1,1,3,1,1,3,46,3,1,1,3,1,1,1,22,1,1,1,3,1,1,3,46,3,1,1,3,1,1,1
%N A040518 Continued fraction for sqrt(542).
%H A040518 Index entries for continued fractions for constants
%p A040518 with(numtheory): Digits:=300: convert(evalf(sqrt(542)),confrac);
%K A040518 nonn,cofr,easy
%O A040518 0,1
%A A040518 njas
%I A040517
%S A040517 23,3,1,5,1,8,2,4,1,2,3,1,1,11,15,2,2,1,1,1,1,1,1,2,2,15,11,1,1,
%T A040517 3,2,1,4,2,8,1,5,1,3,46,3,1,5,1,8,2,4,1,2,3,1,1,11,15,2,2,1,1,1,
%U A040517 1,1,1,2,2,15,11,1,1,3,2,1,4,2,8,1,5,1,3,46,3,1,5,1,8,2,4,1,2,3
%N A040517 Continued fraction for sqrt(541).
%H A040517 Index entries for continued fractions for constants
%p A040517 with(numtheory): Digits:=300: convert(evalf(sqrt(541)),confrac);
%K A040517 nonn,cofr,easy
%O A040517 0,1
%A A040517 njas
%I A040519
%S A040519 23,3,3,3,1,14,1,3,3,3,46,3,3,3,1,14,1,3,3,3,46,3,3,3,1,14,1,3,
%T A040519 3,3,46,3,3,3,1,14,1,3,3,3,46,3,3,3,1,14,1,3,3,3,46,3,3,3,1,14,
%U A040519 1,3,3,3,46,3,3,3,1,14,1,3,3,3,46,3,3,3,1,14,1,3,3,3,46,3,3,3,1
%N A040519 Continued fraction for sqrt(543).
%H A040519 Index entries for continued fractions for constants
%p A040519 with(numtheory): Digits:=300: convert(evalf(sqrt(543)),confrac);
%K A040519 nonn,cofr,easy
%O A040519 0,1
%A A040519 njas
%I A040520
%S A040520 23,3,11,3,46,3,11,3,46,3,11,3,46,3,11,3,46,3,11,3,46,3,11,3,46,
%T A040520 3,11,3,46,3,11,3,46,3,11,3,46,3,11,3,46,3,11,3,46,3,11,3,46,3,
%U A040520 11,3,46,3,11,3,46,3,11,3,46,3,11,3,46,3,11,3,46,3,11,3,46,3,11
%N A040520 Continued fraction for sqrt(544).
%H A040520 Index entries for continued fractions for constants
%p A040520 with(numtheory): Digits:=300: convert(evalf(sqrt(544)),confrac);
%K A040520 nonn,cofr,easy
%O A040520 0,1
%A A040520 njas
%I A040515
%S A040515 23,4,1,1,1,1,1,4,46,4,1,1,1,1,1,4,46,4,1,1,1,1,1,4,46,4,1,1,1,
%T A040515 1,1,4,46,4,1,1,1,1,1,4,46,4,1,1,1,1,1,4,46,4,1,1,1,1,1,4,46,4,
%U A040515 1,1,1,1,1,4,46,4,1,1,1,1,1,4,46,4,1,1,1,1,1,4,46,4,1,1,1,1,1,4
%N A040515 Continued fraction for sqrt(539).
%H A040515 Index entries for continued fractions for constants
%p A040515 with(numtheory): Digits:=300: convert(evalf(sqrt(539)),confrac);
%K A040515 nonn,cofr,easy
%O A040515 0,1
%A A040515 njas
%I A040516
%S A040516 23,4,4,1,10,1,4,4,46,4,4,1,10,1,4,4,46,4,4,1,10,1,4,4,46,4,4,1,
%T A040516 10,1,4,4,46,4,4,1,10,1,4,4,46,4,4,1,10,1,4,4,46,4,4,1,10,1,4,4,
%U A040516 46,4,4,1,10,1,4,4,46,4,4,1,10,1,4,4,46,4,4,1,10,1,4,4,46,4,4,1
%N A040516 Continued fraction for sqrt(540).
%H A040516 Index entries for continued fractions for constants
%p A040516 with(numtheory): Digits:=300: convert(evalf(sqrt(540)),confrac);
%K A040516 nonn,cofr,easy
%O A040516 0,1
%A A040516 njas
%I A040513
%S A040513 23,5,1,3,2,1,1,1,2,1,14,1,2,1,1,1,2,3,1,5,46,5,1,3,2,1,1,1,2,1,
%T A040513 14,1,2,1,1,1,2,3,1,5,46,5,1,3,2,1,1,1,2,1,14,1,2,1,1,1,2,3,1,5,
%U A040513 46,5,1,3,2,1,1,1,2,1,14,1,2,1,1,1,2,3,1,5,46,5,1,3,2,1,1,1,2,1
%N A040513 Continued fraction for sqrt(537).
%H A040513 Index entries for continued fractions for constants
%p A040513 with(numtheory): Digits:=300: convert(evalf(sqrt(537)),confrac);
%K A040513 nonn,cofr,easy
%O A040513 0,1
%A A040513 njas
%I A040514
%S A040514 23,5,7,1,1,7,5,46,5,7,1,1,7,5,46,5,7,1,1,7,5,46,5,7,1,1,7,5,46,
%T A040514 5,7,1,1,7,5,46,5,7,1,1,7,5,46,5,7,1,1,7,5,46,5,7,1,1,7,5,46,5,
%U A040514 7,1,1,7,5,46,5,7,1,1,7,5,46,5,7,1,1,7,5,46,5,7,1,1,7,5,46,5,7
%N A040514 Continued fraction for sqrt(538).
%H A040514 Index entries for continued fractions for constants
%p A040514 with(numtheory): Digits:=300: convert(evalf(sqrt(538)),confrac);
%K A040514 nonn,cofr,easy
%O A040514 0,1
%A A040514 njas
%I A040512
%S A040512 23,6,1,1,2,5,2,1,1,6,46,6,1,1,2,5,2,1,1,6,46,6,1,1,2,5,2,1,1,6,
%T A040512 46,6,1,1,2,5,2,1,1,6,46,6,1,1,2,5,2,1,1,6,46,6,1,1,2,5,2,1,1,6,
%U A040512 46,6,1,1,2,5,2,1,1,6,46,6,1,1,2,5,2,1,1,6,46,6,1,1,2,5,2,1,1,6
%N A040512 Continued fraction for sqrt(536).
%H A040512 Index entries for continued fractions for constants
%p A040512 with(numtheory): Digits:=300: convert(evalf(sqrt(536)),confrac);
%K A040512 nonn,cofr,easy
%O A040512 0,1
%A A040512 njas
%I A040511
%S A040511 23,7,1,2,4,1,3,1,4,2,1,7,46,7,1,2,4,1,3,1,4,2,1,7,46,7,1,2,4,1,
%T A040511 3,1,4,2,1,7,46,7,1,2,4,1,3,1,4,2,1,7,46,7,1,2,4,1,3,1,4,2,1,7,
%U A040511 46,7,1,2,4,1,3,1,4,2,1,7,46,7,1,2,4,1,3,1,4,2,1,7,46,7,1,2,4,1
%N A040511 Continued fraction for sqrt(535).
%H A040511 Index entries for continued fractions for constants
%p A040511 with(numtheory): Digits:=300: convert(evalf(sqrt(535)),confrac);
%K A040511 nonn,cofr,easy
%O A040511 0,1
%A A040511 njas
%I A058287
%S A058287 23,7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,1,4,1,2,108,2,2,1,3,1,7,1,2,2,
%T A058287 2,1,2,3,2,166,1,2,1,4,8,10,1,1,7,1,2,3,566,1,2,3,3,1,20,1,2,19,1,3,2,
%U A058287 1,2,13,2,2,11,3,1,2,1,7,2,1,1,1,2,1,19,1,1,12,11,1,4,1,6,1,2,18,1,2
%N A058287 Continued fraction for e^Pi.
%C A058287 "The transcendentality of e^{Pi} was proved in 1929." (Wells)
%D A058287 Jan Gullberg, "Mathematics, From the Birth of Numbers," W.W. Norton and Company, NY and London, 1997, page 86.
%D A058287 David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 81.
%H A058287 Index entries for continued fractions for constants
%p A058287 with(numtheory): cfrac(evalf((exp(1))^(evalf(Pi)),2560),256,'quotients');
%t A058287 ContinuedFraction[ E^Pi, 100]
%o A058287 (PARI.2.0.17) \p 300 contfrac(exp(1)^Pi)
%K A058287 cofr,nonn,easy
%O A058287 0,1
%A A058287 Robert G. Wilson v (rgwv@kspaint.com), Dec 07 2000
%E A058287 More terms from Jason Earls (jcearls@kskc.net), Jun 21 2001
%I A040510
%S A040510 23,9,4,1,1,22,1,1,4,9,46,9,4,1,1,22,1,1,4,9,46,9,4,1,1,22,1,1,
%T A040510 4,9,46,9,4,1,1,22,1,1,4,9,46,9,4,1,1,22,1,1,4,9,46,9,4,1,1,22,
%U A040510 1,1,4,9,46,9,4,1,1,22,1,1,4,9,46,9,4,1,1,22,1,1,4,9,46,9,4,1,1
%N A040510 Continued fraction for sqrt(534).
%H A040510 Index entries for continued fractions for constants
%p A040510 with(numtheory): Digits:=300: convert(evalf(sqrt(534)),confrac);
%K A040510 nonn,cofr,easy
%O A040510 0,1
%A A040510 njas
%I A013360
%S A013360 1,0,0,1,0,23,10,719,1288,40039,152934,3522539,21952084,447600515,
%T A013360 4051623706,77667878287,956802094896,17678180216111,283850233488142,
%U A013360 5114563630418963,103778358157367212,1833901052655286363
%V A013360 1,0,0,1,0,-23,10,719,-1288,-40039,152934,3522539,-21952084,
%W A013360 -447600515,4051623706,77667878287,-956802094896,-17678180216111,
%X A013360 283850233488142,5114563630418963,-103778358157367212
%N A013360 exp(sin(x)-arctan(x))=1+1/3!*x^3-23/5!*x^5+10/6!*x^6+719/7!*x^7...
%K A013360 sign,done
%O A013360 0,6
%A A013360 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A013491
%S A013491 1,0,0,1,0,23,10,719,1288,40599,152934,3735059,21982884,510402683,
%T A013491 4077409466,96691506911,972083601136,24171783139087,292315200462606,
%U A013491 7694747756773355,108645176948603484,3036089243696854595
%V A013491 1,0,0,-1,0,-23,10,-719,1288,-40599,152934,-3735059,21982884,
%W A013491 -510402683,4077409466,-96691506911,972083601136,-24171783139087,
%X A013491 292315200462606,-7694747756773355,108645176948603484
%N A013491 exp(sinh(x)-arctanh(x))=1-1/3!*x^3-23/5!*x^5+10/6!*x^6-719/7!*x^7...
%K A013491 sign,done
%O A013491 0,6
%A A013491 Patrick Demichel (dml@hpfrcu03.france.hp.com)
%I A040509
%S A040509 23,11,1,1,11,46,11,1,1,11,46,11,1,1,11,46,11,1,1,11,46,11,1,1,
%T A040509 11,46,11,1,1,11,46,11,1,1,11,46,11,1,1,11,46,11,1,1,11,46,11,1,
%U A040509 1,11,46,11,1,1,11,46,11,1,1,11,46,11,1,1,11,46,11,1,1,11,46,11
%N A040509 Continued fraction for sqrt(533).
%H A040509 Index entries for continued fractions for constants
%p A040509 with(numtheory): Digits:=300: convert(evalf(sqrt(533)),confrac);
%K A040509 nonn,cofr,easy
%O A040509 0,1
%A A040509 njas
%I A033343
%S A033343 23,11,7,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,
%T A033343 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A033343 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A033343 [ 23/n ].
%K A033343 easy,nonn
%O A033343 1,1
%A A033343 Jeff Burch (jmburch@osprey.smcm.edu)
%I A054574
%S A054574 23,11,17,23,47,41,53,59,71,89,167,113,269,131,167,191,179,227,239,263,
%T A054574 251,239,251,269,293,431,311,359,383,383,383,479,479,419,449,881,2039,
%U A054574 491,503,521,2039,659,2039,743,593,599,839,743,683,911,701,719,1103
%N A054574 Begin with n-th prime, add its prime divisors (itself), repeat until reach a new prime; sequence gives prime reached.
%e A054574 a(5)=47, because starting with the 5th prime, 11: 11+11=22; 22+2+11=35; 35+5+7=47, a prime.
%Y A054574 Cf. A054575.
%K A054574 easy,nonn
%O A054574 1,1
%A A054574 Enoch Haga (EnochHaga@msn.com), Apr 11 2000
%E A054574 Corrected by Jud McCranie (jud.mccranie@mindspring.com), Jan 04 2001
%I A002549 M5128 N2223
%S A002549 1,1,23,11,563,1627,88069,1423,1593269,7759469,31730711,46522243
%N A002549 Numerators of coefficients for numerical differentiation.
%D A002549 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).
%Y A002549 Cf. A002550.
%K A002549 nonn
%O A002549 1,3
%A A002549 njas
%I A064735
%S A064735 23,13,53,17,113,113,173,191,223,229,131,137,241,431,347,353,359,461,
%T A064735 167,271,173,179,283,389,197,1013,1031,5107,1091,2113,1277,1319,1373,
%U A064735 1399,1493,1151,1571,1163,3167,1733,2179,1181,1913,1193,1973,1993,2111
%N A064735 Next prime containing prime(n) in decimal notation
%Y A064735 A062584, A030670, A000040.
%K A064735 base,nonn
%O A064735 1,1
%A A064735 Reinhard Zumkeller (reinhard.zumkeller@lhsystems.com), Oct 17 2001
%I A040508
%S A040508 23,15,2,1,4,2,4,1,2,15,46,15,2,1,4,2,4,1,2,15,46,15,2,1,4,2,4,
%T A040508 1,2,15,46,15,2,1,4,2,4,1,2,15,46,15,2,1,4,2,4,1,2,15,46,15,2,1,
%U A040508 4,2,4,1,2,15,46,15,2,1,4,2,4,1,2,15,46,15,2,1,4,2,4,1,2,15,46
%N A040508 Continued fraction for sqrt(532).
%H A040508 Index entries for continued fractions for constants
%p A040508 with(numtheory): Digits:=300: convert(evalf(sqrt(532)),confrac);
%K A040508 nonn,cofr,easy
%O A040508 0,1
%A A040508 njas
%I A004512
%S A004512 23,21,22,26,24,25,20,18,19,5,3,4,8,6,7,2,0,1,14,12,13,
%T A004512 17,15,16,11,9,10,50,48,49,53,51,52,47,45,46,32,30,31,35,
%U A004512 33,34,29,27,28,41,39,40,44,42,43,38,36,37,77,75,76,80
%N A004512 Tersum n + 23.
%F A004512 Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g. 5 + 8 = "21" + "22" = "10" = 1.
%K A004512 nonn
%O A004512 0,1
%A A004512 njas
%I A022979
%S A022979 23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,
%T A022979 3,2,1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,
%U A022979 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37
%V A022979 23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,
%W A022979 3,2,1,0,-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,-11,-12,-13,-14,-15,-16,-17,-18,-19,
%X A022979 -20,-21,-22,-23,-24,-25,-26,-27,-28,-29,-30,-31,-32,-33,-34,-35,-36,-37
%N A022979 23-n.
%K A022979 sign,done
%O A022979 0,1
%A A022979 njas
%I A023465
%S A023465 23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,
%T A023465 3,2,1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,
%U A023465 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37
%V A023465 -23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,
%W A023465 -3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,
%X A023465 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37
%N A023465 n-23.
%K A023465 sign,done
%O A023465 0,1
%A A023465 njas
%I A004464
%S A004464 23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,
%T A004464 4,3,2,1,0,15,14,13,12,11,10,9,8,55,54,53,52,51,50,49,48,
%U A004464 63,62,61,60,59,58,57,56,39,38,37,36,35,34,33,32,47,46
%N A004464 Nimsum n + 23.
%D A004464 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
%D A004464 J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
%H A004464 Index entries for sequences related to Nim-sums
%K A004464 nonn
%O A004464 0,1
%A A004464 njas
%I A010862
%S A010862 23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,
%T A010862 23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,
%U A010862 23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23
%N A010862 Constant sequence.
%K A010862 nonn
%O A010862 0,1
%A A010862 njas
%I A040507
%S A040507 23,23,46,23,46,23,46,23,46,23,46,23,46,23,46,23,46,23,46,23,46,
%T A040507 23,46,23,46,23,46,23,46,23,46,23,46,23,46,23,46,23,46,23,46,23,
%U A040507 46,23,46,23,46,23,46,23,46,23,46,23,46,23,46,23,46,23,46,23,46
%N A040507 Continued fraction for sqrt(531).
%H A040507 Index entries for continued fractions for constants
%p A040507 with(numtheory): Digits:=300: convert(evalf(sqrt(531)),confrac);
%K A040507 nonn,cofr,easy
%O A040507 0,1
%A A040507 njas
%I A022357
%S A022357 0,23,23,46,69,115,184,299,483,782,1265,2047,3312,5359,
%T A022357 8671,14030,22701,36731,59432,96163,155595,251758,407353,
%U A022357 659111,1066464,1725575,2792039,4517614,7309653,11827267
%N A022357 Fibonacci sequence beginning 0 23.
%K A022357 nonn
%O A022357 0,2
%A A022357 njas
%I A062999
%S A062999 23,24,25,26,27,28,29,32,33,34,35,36,37,38,39,42,43,44,45,46,47,48,49,
%T A062999 52,53,54,55,56,57,58,59,62,63,64,65,66,67,68,69,72,73,74,75,76,77,78,
%U A062999 79,82,83,84,85,86,87,88,89,92,93,94,95,96,97,98,99,124,125,126,127
%N A062999 Sum of digits is strictly less than product of digits.
%Y A062999 Cf. A007953, A007954, A034710, A062329, A062996, A062997, A062998, A062999.
%K A062999 base,nonn
%O A062999 1,1
%A A062999 Henry Bottomley (se16@btinternet.com), Jun 29 2001
%I A007638 M5129
%S A007638 23,24,28,31,39,44,45,46,47,50,52,56,57,60,63,67,69,70,71,79,80,85,86,
%T A007638 88,89,90,92,93,96,97,102,107,108,112,115,116,118,119,121,122,123,126,
%U A007638 128,131,134,137,138,139,143,144,145,147,148,151,153,156,157,161,162
%N A007638 3n^2 - 3n + 23 is composite.
%K A007638 nonn
%O A007638 1,1
%A A007638 njas, Mira Bernstein, Robert G. Wilson v (rgwv@kspaint.com)
%I A031332
%S A031332 23,24,30,34,44,53,54,61,67,86,111,121,124,129,133,149,152,189,
%T A031332 204,205,211,222,236,242,247,252,254,255,256,258,284,306,312,
%U A031332 314,317,340,342,361,365,370,386,396,399,409,414,421,423,441
%N A031332 Position of n-th 7 in A031324.
%K A031332 nonn
%O A031332 1,1
%A A031332 Clark Kimberling, ck6@cedar.evansville.edu
%I A022393
%S A022393 1,23,24,47,71,118,189,307,496,803,1299,2102,3401,5503,
%T A022393 8904,14407,23311,37718,61029,98747,159776,258523,418299,
%U A022393 676822,1095121,1771943,2867064,4639007,7506071,12145078
%N A022393 Fibonacci sequence beginning 1 23.
%K A022393 nonn
%O A022393 0,2
%A A022393 njas
%I A042068
%S A042068 23,24,47,71,118,189,874,1063,1937,3000,4937,7937,370039,
%T A042068 377976,748015,1125991,1874006,2999997,13873994,16873991,
%U A042068 30747985,47621976,78369961,125991937,5873999063,5999991000
%N A042068 Numerators of continued fraction convergents to sqrt(558).
%Y A042068 Cf. A042069.
%K A042068 nonn,cofr,easy
%O A042068 0,1
%A A042068 njas
%I A042066
%S A042066 23,24,47,71,118,5499,5617,11116,16733,27849,1297787,1325636,
%T A042066 2623423,3949059,6572482,306283231,312855713,619138944,
%U A042066 931994657,1551133601,72284140303,73835273904,146119414207
%N A042066 Numerators of continued fraction convergents to sqrt(557).
%Y A042066 Cf. A042067.
%K A042066 nonn,cofr,easy
%O A042066 0,1
%A A042066 njas
%I A042070
%S A042070 23,24,47,71,331,5036,5367,10403,46979,57382,161743,219125,
%T A042070 1038243,1257368,2295611,35691533,145061743,180753276,325815019,
%U A042070 506568295,23627956589,24134524884,47762481473,71897006357
%N A042070 Numerators of continued fraction convergents to sqrt(559).
%Y A042070 Cf. A042071.
%K A042070 nonn,cofr,easy
%O A042070 0,1
%A A042070 njas
%I A042072
%S A042072 23,24,47,71,3313,3384,6697,10081,470423,480504,950927,
%T A042072 1431431,66796753,68228184,135024937,203253121,9484668503,
%U A042072 9687921624,19172590127,28860511751,1346756130673,1375616642424
%N A042072 Numerators of continued fraction convergents to sqrt(560).
%Y A042072 Cf. A042073.
%K A042072 nonn,cofr,easy
%O A042072 0,1
%A A042072 njas
%I A042064
%S A042064 23,24,47,118,165,283,448,1627,5329,12285,17614,100355,
%T A042064 519389,7891190,8410579,16301769,138824731,155126500,1690089731,
%U A042064 1845216231,16451819579,18297035810,34748855389,539529866645
%N A042064 Numerators of continued fraction convergents to sqrt(556).
%Y A042064 Cf. A042065.
%K A042064 nonn,cofr,easy
%O A042064 0,1
%A A042064 njas
%I A042062
%S A042062 23,24,47,165,212,801,1013,1814,84457,86271,170728,598455,
%T A042062 769183,2906004,3675187,6581191,306409973,312991164,619401137,
%U A042062 2171194575,2790595712,10542981711,13333577423,23876559134
%N A042062 Numerators of continued fraction convergents to sqrt(555).
%Y A042062 Cf. A042063.
%K A042062 nonn,cofr,easy
%O A042062 0,1
%A A042062 njas
%I A042060
%S A042060 23,24,47,306,1271,1577,2848,12969,80662,93631,174293,8111109,
%T A042060 8285402,16396511,106664468,443054383,549718851,992773234,
%U A042060 4520811787,28117643956,32638455743,60756099699,2827419041897
%N A042060 Numerators of continued fraction convergents to sqrt(554).
%Y A042060 Cf. A042061.
%K A042060 nonn,cofr,easy
%O A042060 0,1
%A A042060 njas
%I A042058
%S A042058 23,24,47,729,3692,4421,21376,47173,68549,115722,415715,
%T A042058 531437,1478589,9402971,20284531,29687502,109347037,139034539,
%U A042058 248381576,635797691,2791572340,3427370031,19928422495
%N A042058 Numerators of continued fraction convergents to sqrt(553).
%Y A042058 Cf. A042059.
%K A042058 nonn,cofr,easy
%O A042058 0,1
%A A042058 njas
%I A045859
%S A045859 23,24,71,72,73,74,75,76,77,224,225,226,227,228,229,230,231,232,233,
%T A045859 234,235,236,237,238,239,240,241,242,243,244,708,709,710,711,712,713,
%U A045859 714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730
%N A045859 n^2 has initial digit '5'.
%K A045859 nonn
%O A045859 0,1
%A A045859 Jeff Burch (gburch@erols.com)
%I A042078
%S A042078 23,24,71,95,261,6098,12457,18555,49567,68122,3183179,3251301,
%T A042078 9685781,12937082,35559945,830815817,1697191579,2528007396,
%U A042078 6753206371,9281213767,433689039653,442970253420,1319629546493
%N A042078 Numerators of continued fraction convergents to sqrt(563).
%Y A042078 Cf. A042079.
%K A042078 nonn,cofr,easy
%O A042078 0,1
%A A042078 njas
%I A042080
%S A042080 23,24,71,95,4441,4536,13513,18049,843767,861816,2567399,
%T A042080 3429215,160311289,163740504,487792297,651532801,30458301143,
%U A042080 31109833944,92677969031,123787802975,5786916905881,5910704708856
%N A042080 Numerators of continued fraction convergents to sqrt(564).
%Y A042080 Cf. A042081.
%K A042080 nonn,cofr,easy
%O A042080 0,1
%A A042080 njas
%I A042076
%S A042076 23,24,71,166,403,1778,2181,12683,14864,339691,354555,2112466,
%T A042076 2467021,11980550,26428121,64836792,156101705,220938497,
%U A042076 10319272567,10540211064,31399694695,73339600454,178078895603
%N A042076 Numerators of continued fraction convergents to sqrt(562).
%Y A042076 Cf. A042077.
%K A042076 nonn,cofr,easy
%O A042076 0,1
%A A042076 njas
%I A042074
%S A042074 23,24,71,379,450,829,2108,5045,12198,17243,29441,164448,
%T A042074 358337,522785,24406447,24929232,74264911,396253787,470518698,
%U A042074 866772485,2204063668,5274899821,12753863310,18028763131
%N A042074 Numerators of continued fraction convergents to sqrt(561).
%Y A042074 Cf. A042075.
%K A042074 nonn,cofr,easy
%O A042074 0,1
%A A042074 njas
%I A042084
%S A042084 23,24,95,119,452,571,1023,8755,9778,223871,233649,2093063,
%T A042084 2326712,4419775,15586037,20005812,75603473,95609285,4473630583,
%U A042084 4569239868,18181350187,22750590055,86433120352,109183710407
%N A042084 Numerators of continued fraction convergents to sqrt(566).
%Y A042084 Cf. A042085.
%K A042084 nonn,cofr,easy
%O A042084 0,1
%A A042084 njas
%I A042082
%S A042082 23,24,95,214,309,3613,18374,95483,1068687,1164170,3397027,
%T A042082 11355251,14752278,689960039,704712317,2804096990,6312906297,
%U A042082 9117003287,106599942454,542116715557,2817183520239,31531135438186
%N A042082 Numerators of continued fraction convergents to sqrt(565).
%Y A042082 Cf. A042083.
%K A042082 nonn,cofr,easy
%O A042082 0,1
%A A042082 njas
%I A042088
%S A042088 23,24,119,143,6697,6840,34057,40897,1915319,1956216,9740183,
%T A042088 11696399,547774537,559470936,2785658281,3345129217,156661602263,
%U A042088 160006731480,796688528183,956695259663,44804670472681
%N A042088 Numerators of continued fraction convergents to sqrt(568).
%Y A042088 Cf. A042089.
%K A042088 nonn,cofr,easy
%O A042088 0,1
%A A042088 njas
%I A042086
%S A042086 23,24,119,381,1643,2024,94747,96771,481831,1542264,6650887,
%T A042086 8193151,383535833,391728984,1950451769,6243084291,26922788933,
%U A042086 33165873224,1552552957237,1585718830461,7895428279081
%N A042086 Numerators of continued fraction convergents to sqrt(567).
%Y A042086 Cf. A042087.
%K A042086 nonn,cofr,easy
%O A042086 0,1
%A A042086 njas
%I A042090
%S A042090 23,24,143,167,978,8969,18916,27885,74686,251943,830515,
%T A042090 1912973,2743488,7399949,69343029,354115094,423458123,2471405709,
%U A042090 2894863832,135635141981,138530005813,828285171046,966815176859
%N A042090 Numerators of continued fraction convergents to sqrt(569).
%Y A042090 Cf. A042091.
%K A042090 nonn,cofr,easy
%O A042090 0,1
%A A042090 njas
%I A042092
%S A042092 23,24,167,191,8953,9144,63817,72961,3420023,3492984,24377927,
%T A042092 27870911,1306439833,1334310744,9312304297,10646615041,
%U A042092 499056596183,509703211224,3557275863527,4066979074751
%N A042092 Numerators of continued fraction convergents to sqrt(570).
%Y A042092 Cf. A042093.
%K A042092 nonn,cofr,easy
%O A042092 0,1
%A A042092 njas
%I A042094
%S A042094 23,24,215,239,454,1147,1601,2748,4349,67983,208298,484579,
%T A042094 692877,4641841,33185764,37827605,184496184,406819973,1404956103,
%U A042094 6026644385,13458244873,315566276464,644590797801,2893929467668
%N A042094 Numerators of continued fraction convergents to sqrt(571).
%Y A042094 Cf. A042095.
%K A042094 nonn,cofr,easy
%O A042094 0,1
%A A042094 njas
%I A042096
%S A042096 23,24,263,287,13465,13752,150985,164737,7728887,7893624,
%T A042096 86665127,94558751,4436367673,4530926424,49745631913,54276558337,
%U A042096 2546467315415,2600743873752,28553906052935,31154649926687
%N A042096 Numerators of continued fraction convergents to sqrt(572).
%Y A042096 Cf. A042097.
%K A042096 nonn,cofr,easy
%O A042096 0,1
%A A042096 njas
%I A042098
%S A042098 23,24,359,383,17977,18360,275017,293377,13770359,14063736,
%T A042098 210662663,224726399,10548077017,10772803416,161367324841,
%U A042098 172140128257,8079813224663,8251953352920,123607160165543
%N A042098 Numerators of continued fraction convergents to sqrt(573).
%Y A042098 Cf. A042099.
%K A042098 nonn,cofr,easy
%O A042098 0,1
%A A042098 njas
%I A041263
%S A041263 1,1,23,24,551,575,13201,13776,316273,330049,7577351,7907400,
%T A041263 181540151,189447551,4349386273,4538833824,104203730401,
%U A041263 108742564225,2496540143351,2605282707576,59812759710023
%N A041263 Denominators of continued fraction convergents to sqrt(143).
%Y A041263 Cf. A041262.
%K A041263 nonn,cofr,easy
%O A041263 0,3
%A A041263 njas
%I A042100
%S A042100 23,24,551,575,27001,27576,633673,661249,31051127,31712376,
%T A042100 728723399,760435775,35708769049,36469204824,838031275177,
%U A042100 874500480001,41065053355223,41939553835224,963735237730151
%N A042100 Numerators of continued fraction convergents to sqrt(574).
%Y A042100 Cf. A042101.
%K A042100 nonn,cofr,easy
%O A042100 0,1
%A A042100 njas
%I A042101
%S A042101 1,1,23,24,1127,1151,26449,27600,1296049,1323649,30416327,
%T A042101 31739976,1490455223,1522195199,34978749601,36500944800,
%U A042101 1714022210401,1750523155201,40225531624823,41976054780024
%N A042101 Denominators of continued fraction convergents to sqrt(574).
%Y A042101 Cf. A042100.
%K A042101 nonn,cofr,easy
%O A042101 0,3
%A A042101 njas
%I A042102
%S A042102 23,24,1127,1151,54073,55224,2594377,2649601,124476023,
%T A042102 127125624,5972254727,6099380351,286543750873,292643131224,
%U A042102 13748127787177,14040770918401,659623590033623,673664360952024
%N A042102 Numerators of continued fraction convergents to sqrt(575).
%Y A042102 Cf. A042103.
%K A042102 nonn,cofr,easy
%O A042102 0,1
%A A042102 njas
%I A054865
%S A054865 23,25,29,31,32,34,35,41,43,47,49,51,52,53,58,59,61,62,65,67,68,69,71,
%T A054865 77,79,167,169,187,191,193,197,203,205,207,209,211,213,221,223,225,227,
%U A054865 229,233,239,241,245,247,248,250,251,263,265,269,274,277,278,281,283
%N A054865 Numbers not divisible by any of their digits when written in base 9.
%Y A054865 Cf. A038772, A055239, A055240.
%K A054865 base,easy,nonn
%O A054865 1,1
%A A054865 Henry Bottomley (se16@btinternet.com), May 12 2000
%I A054795
%S A054795 23,25,29,31,35,41,43,53,55,65,73,79,95,113,121,133,151,155,179,185,
%T A054795 205,223,233,263,265,295,311,329,361,365,403,415,443,473,485,529,535,
%U A054795 575,601,623,671,673,725,745,779,823,835,893,905,953,991,1015,1079
%N A054795 Numbers of form 23+n^2+n or 23+2*n^2.
%Y A054795 Cf. A052293, A054794, A054796.
%K A054795 nonn
%O A054795 1,1
%A A054795 stuart m. ellerstein (ellerstein@aol.com), Apr 27 2000
%E A054795 More terms from James A. Sellers (sellersj@math.psu.edu), Apr 28 2000
%I A033629
%S A033629 23,25,33,35,43,45,67,92,94,96,111,121,136,143,160,165,170,172,187,194,
%T A033629 204,226,231,248,265,270,280,287,292,297,302,304,314,331,336,346,348,
%U A033629 353,368,380,397,407,419,424,446,463,468,473,475,480,490,495,507
%N A033629 Not the sum of two distinct Ulam numbers.
%D A033629 R. K. Guy, Unsolved Problems in Number Theory, C4
%Y A033629 Cf. A002858.
%K A033629 nonn
%O A033629 1,1
%A A033629 Jud McCranie (jud.mccranie@mindspring.com)
%I A050864
%S A050864 1,23,25,53,65,139,149,481,613,1343,1601,2125,4385,9005,9169
%N A050864 225*2^n-1 is prime.
%H A050864 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime
%H A050864 R. Ballinger and W. Keller, List of primes k.2^n + 1 for k < 300
%H A050864 R. Ballinger and W. Keller, List of primes k.2^n - 1 for k < 300
%K A050864 hard,nonn
%O A050864 0,2
%A A050864 njas, Dec 29 1999
%I A025059
%S A025059 23,26,29,31,35,38,39,41,44,47,50,51,53,54,55,56,59,61,62,63,65,66,68,69,71,
%T A025059 74,75,76,77,79,80,81,83,84,86,87,89,90,91,92,94,95,96,98,99,101,103,104,
%U A025059 106,107,108,109,110,111,113,114,115,116,117,118,119,121,122,123,124,125
%N A025059 Numbers expressible in more than one way as j*k + k*i + i*j, where 1 <=i < j < k.
%K A025059 nonn
%O A025059 1,1
%A A025059 Clark Kimberling (ck6@cedar.evansville.edu)
%I A038772
%S A038772 23,27,29,34,37,38,43,46,47,49,53,54,56,57,58,59,67,68,69,73,74,76,78,
%T A038772 79,83,86,87,89,94,97,98,203,207,209,223,227,229,233,239,247,249,253,
%U A038772 257,259,263,267,269,277,283,289,293,299,307,308,323,329,334,337,338
%N A038772 Numbers not divisible by any of their digits.
%e A038772 35 is excluded because 5 is a divisor of 35, but 37 is included because neither 3 nor 7 is a divisor of 37
%Y A038772 Cf. A034709, A034837, A038769, A038770.
%K A038772 base,easy,nonn,nice
%O A038772 1,1
%A A038772 Henry Bottomley (se16@btinternet.com), May 04 2000
%I A045806
%S A045806 23,27,61,89,123,127,161,189,223,227,261,289,323,327,361,389,423,427,
%T A045806 461,489,523,527,561,589,623,627,661,689,723,727,761,789,823,827,861,
%U A045806 889,923,927,961,989,1023,1027,1061,1089,1123,1127,1161,1189,1223,1227
%N A045806 6-ish numbers (end in 23, 27, 61, 89).
%Y A045806 Cf. A045800-A045809.
%K A045806 nonn,base,easy
%O A045806 0,1
%A A045806 J. H. Conway.
%E A045806 More terms from Erich Friedman (erich.friedman@stetson.edu).
%I A060703
%S A060703 23,28,56,69,84,92,119,140,161,224,237,238,253,359,364,414,474,476,588,
%T A060703 595,667,711,796,833,839,952,1016,1071,1077,1081,1185,1428,1540,1666,
%U A060703 1679,1748,1795,1896,1918,2032,2154,2261,2388,2492,2513,2737,2829
%N A060703 Negative values of 2*x*y^4+x^2*y^3-2*x^3*y*2-x^4*y-y^5+2*y for x,y >= 0.
%C A060703 Positive values of the polynomial for x,y >= 0 are the Fibonacci numbers. See A000045. Ribenboim discusses the equation on page 193.
%D A060703 Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pg 193, 1996.
%e A060703 For x=2, y=1, the value of the polynomial is -23, so 23 is in the sequence.
%Y A060703 Cf. A000045.
%K A060703 nonn
%O A060703 0,1
%A A060703 Jud McCranie (jud.mccranie@mindspring.com), Apr 20 2001
%I A061753
%S A061753 23,29,31,35,39,41,44,47,49,53,55,59,62,63,65,69,71,74,76,77,79,80,83,
%T A061753 87,89,90,95,97,98,99,101,103,104,107,109,111,113,116,118,119,120,124,
%U A061753 125,127,129,131,132,134,135,137,139,142,143,146,149
%N A061753 n! is divisible by (n+1)^5.
%t A061753 Select[Range[150], IntegerQ[ #!/(# + 1)^5] &]
%K A061753 nonn
%O A061753 1,1
%A A061753 Robert G. Wilson v (rgwv@kspaint.com), Jun 21 2001
%I A049483
%S A049483 23,29,31,37,41,47,61,67,71,73,79,83,89,101,107,113,127,131,137,149,
%T A049483 157,163,167,193,211,229,233,239,241,269,281,283,307,311,337,347,349,
%U A049483 353,367,373,379,383,389,397,401,409,419,421,431,439,443,457,467,479
%N A049483 Both p and p+Q(5) are primes, where Q(5)=2310 is the 5th primorial number (A002110[ 5 ]).
%C A049483 p and p+2310 are not necessarily consecutive primes
%e A049483 23 and 23 + 2*3*5*7*11 = 2333 is prime
%Y A049483 A045320, A001359, A023201, sequ for p+30, seq for 210
%K A049483 nonn
%O A049483 1,1
%A A049483 Labos E. (labos@ana1.sote.hu)
%I A061754
%S A061754 23,29,35,39,44,47,53,55,59,62,63,69,71,74,76,79,80,83,87,89,90,95,97,
%T A061754 98,99,103,104,107,109,111,116,118,119,124,125,127,129,131,132,134,135,
%U A061754 139,142,143,146,149,151,152,153,155,159,160,161,164,167,168,169,170
%N A061754 n! is divisible by (n+1)^6.
%t A061754 Select[Range[200], IntegerQ[ #!/(# + 1)^6] &]
%K A061754 nonn
%O A061754 1,1
%A A061754 Robert G. Wilson v (rgwv@kspaint.com), Jun 21 2001
%I A007637 M5130
%S A007637 23,29,41,59,83,113,149,191,239,293,353,419,491,569,653,743,839,941,
%T A007637 1049,1163,1283,1409,1823,1973,2129,2459,2633,2999,3191,3389,3593,3803,
%U A007637 4019,4241,4703,4943,5189,5441,6791,7079,7673,8291,8609
%N A007637 Primes of form 3n^2 -3n+23.
%K A007637 nonn
%O A007637 1,1
%A A007637 njas, Mira Bernstein, Robert G. Wilson v (rgwv@kspaint.com)
%I A045120
%S A045120 23,29,53,71,77,83,91,92,94,103,109,113,116,118,121,151,157,181,197,
%T A045120 209,212,214,217,229,263,269,275,283,284,286,295,301,305,308,310,313,
%U A045120 323,331,332,334,355,363,364,366,368,370,376,378
%N A045120 In base 4 representation the numbers of 1's and 3's are 2 and 1, respectively.
%K A045120 nonn,base
%O A045120 1,1
%A A045120 Clark Kimberling, ck6@cedar.evansville.edu
%I A063980
%S A063980 23,29,59,61,67,71,79,83,109,137,139,149,193,227,233,239,251,257,
%T A063980 269,271,277,293,307,311,317,359,379,383,389,397,401,419,431,449,
%U A063980 461,463,467,479,499,503,521,557,563,569,571,577,593,599,601,607
%N A063980 Pillai primes: p such that there exists an integer m such that m!+1 is 0 mod p and p is not 1 mod m.
%D A063980 G E Hardy and M V Subbarao, A modified problem of Pillai and some related questions, preprint, 2001.
%Y A063980 Smallest m is given in A063828.
%K A063980 nonn,nice
%O A063980 1,1
%A A063980 Richard Guy (rkg@cpsc.ucalgary.ca), Sep 08 2001
%E A063980 More terms from David W. Wilson, Sep 08, 2001
%I A046124
%S A046124 23,29,59,79,269,619,659,1109,1499,1619,1759,1879,2389,2689,3319,3929,
%T A046124 4019,5119,5399,5449,5659,6329,6379,9479,11839,12119,12659,13469,
%U A046124 14639,14759,15809,15919,17489,18229,19489,20359,21499,23339,24109
%N A046124 Last member of a sexy prime quadruplet: value of p+18 where (p,p+6,p+12,p+18) are all prime.
%H A046124 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A046124 Cf. A046121, A046122, A046123.
%K A046124 nonn
%O A046124 1,1
%A A046124 Eric W. Weisstein (eric@weisstein.com)
%I A029541
%S A029541 1,23,29,169,575,887,3151,3473,4495,5395,11431,372077
%N A029541 n divides the (left) concatenation of all numbers <= n written in base 24 (most significant digit on right).
%C A029541 This sequence differs from A061977 in that all least significant zeros are removed before concatenation.
%C A029541 The next term is > 400000. - Larry Reeves, Jan 16, 2002
%e A029541 See A029519 for example.
%Y A029541 Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.
%K A029541 nonn,base,more
%O A029541 1,2
%A A029541 Olivier Gerard (ogerard@ext.jussieu.fr)
%E A029541 Additional comments and more terms from Larry Reeves (larryr@acm.org), May 25 2001
%I A036267
%S A036267 23,30,32,28,10,22,48,48,22,6,10,2,14,40,46,20,18,40,36,12,16,20,
%T A036267 4,26,20,28,92,110,86,82,102,100,62,14,16,16,14,62,98,94,70,68,
%U A036267 62,12,28,14,0,26,66,82,60,20,16,42,46,16,28,28,28,58,34,38,86
%V A036267 23,-30,32,-28,10,22,-48,48,-22,-6,10,-2,14,-40,46,-20,-18,40,-36,12,16,-20,
%W A036267 -4,26,-20,-28,92,-110,86,-82,102,-100,62,-14,-16,16,14,-62,98,-94,70,-68,
%X A036267 62,-12,-28,14,0,26,-66,82,-60,20,16,-42,46,-16,-28,28,28,-58,34,-38,86
%N A036267 6th differences of primes.
%K A036267 sign,done
%O A036267 1,1
%A A036267 njas
%I A026051
%S A026051 23,31,40,53,68,86,109,135,167,203,244,292,345,406,473,547,630,720,820,928,1045,
%T A026051 1173,1310,1459,1618,1788,1971,2165,2373,2593,2826,3074,3335,3612,3903,4209,4532,
%U A026051 4870,5226,5598,5987,6395,6820,7265,7728,8210,8713,9235,9779,10343,10928,11536
%N A026051 a(n) = (d(n)-r(n))/5, where d = A026049 and r is the periodic sequence with fund. period (4,1,4,0,1).
%K A026051 nonn
%O A026051 7,1
%A A026051 Clark Kimberling (ck6@cedar.evansville.edu)
%I A060328
%S A060328 23,31,41,59,67,71,109,113,131,139,157,199,211,239,251,269,293,311,337,
%T A060328 379,383,409,419,487,491,499,503,521,571,599,631,701,751,769,773,787,
%U A060328 829,877,881,919,941,953,991,1009,1013,1039,1049,1061,1103,1117,1151
%N A060328 Primes the sum of three consecutive composite numbers.
%e A060328 a(3) = 41 is equal to 12+14+15.
%t A060328 composite[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1 != n, k++ ]; k); b = {}; Do[ p = composite[ n ] + composite[ n + 1 ] + composite[ n + 2 ]; If[ PrimeQ[ p ], b = Append[ b, p ] ], {n, 1, 1000} ]; b
%Y A060328 Cf. A060254.
%K A060328 nonn
%O A060328 1,1
%A A060328 Robert G. Wilson v (rgwv@kspaint.com), Mar 30 2001
%I A034962
%S A034962 23,31,41,59,71,83,97,109,131,173,199,211,223,251,269,311,349,439,457,
%T A034962 487,503,607,661,701,829,857,883,911,941,1033,1049,1061,1151,1187,1229,
%U A034962 1249,1303,1367,1381,1409,1433,1493,1511,1553,1667,1867,1931,1973,1993
%N A034962 Primes that are sum of three consecutive primes.
%e A034962 E.g. 131 = 41 + 43 + 47.
%Y A034962 Cf. A001043, A011974, A034707.
%K A034962 nonn
%O A034962 0,1
%A A034962 Patrick De Geest (pdg@worldofnumbers.com), Oct 1998.
%I A023679
%S A023679 23,31,44,59,76,83,87,104,107,108,116,135,139,140,152,172,175,199,
%T A023679 200,204,211,212,216,231,239,243,244
%N A023679 Discriminants of complex cubic fields.
%D A023679 M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, p. 437.
%K A023679 nonn
%O A023679 1,1
%A A023679 njas
%I A031924
%S A031924 23,31,47,53,61,73,83,131,151,157,167,173,233,251,257,263,271,331,
%T A031924 353,367,373,383,433,443,503,541,557,563,571,587,593,601,607,647,
%U A031924 653,677,727,733,751,941,947,971,977,991,1013,1033,1063,1097,1103
%N A031924 Lower prime of a difference of 6 between consecutive primes.
%e A031924 23 is a term as the next prime 29 = 23 + 6.
%Y A031924 Cf. A031925. A031924 and A007529 together give A023201.
%K A031924 nonn
%O A031924 0,1
%A A031924 Jeff Burch (jmburch@osprey.smcm.edu)
%I A033216
%S A033216 23,31,47,71,89,97,103,113,137,191,199,223,257,311,313,
%T A033216 353,367,383,401,433,449,463,487,521,577,599,617,631,641,
%U A033216 647,719,727,751,823,839,863,881,911,929,977,983,991,1039
%N A033216 Primes of form x^2+22*y^2.
%D A033216 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.
%K A033216 nonn
%O A033216 0,1
%A A033216 njas
%I A054291
%S A054291 23,31,49,58,67,105,112,120,134,156,167,169,173,178,205,215,242,264,
%T A054291 277,282,285,295,298,329,331,338,353,368,388,393,395,408,416,450,453,
%U A054291 469,487,507,516,524,527,539,555
%N A054291 Positions of 5's in the decimal expansion of 1/2*(sqrt(5)+1)).
%K A054291 nonn
%O A054291 0,1
%A A054291 Simon Plouffe (plouffe@math.uqam.ca), Feb 20, 2000
%I A052230
%S A052230 23,31,53,61,83,151,173,233,263,271,331,353,383,443,503,541,563,571,
%T A052230 593,601,653,751,991,1013,1103,1223,1231,1283,1291,1321,1433,1493,1553,
%U A052230 1613,1621,1741,1861,1973,2011,2063,2131,2281,2333,2341,2371,2393,2543
%N A052230 Primes p from A031924 such that A052180(p) = 5.
%Y A052230 A031924, A031925.
%K A052230 nonn
%O A052230 1,1
%A A052230 Labos E. (labos@ana1.sote.hu), Feb 01 2000
%I A064792
%S A064792 23,31,53,71,113,131,173,191,233,293,311,373,419,431,479,5303,593,613,
%T A064792 673,719,733,797,839,8923,971,1013,1031,10709,1091,11311,1277,1319,
%U A064792 1373,1399,1493,1511,1571,1637,16703,1733,17903,1811,1913,1931,1973
%N A064792 Add more digits to p until reach another prime.
%Y A064792 See A030670 for another version. Cf. A065112.
%K A064792 nonn,base
%O A064792 1,1
%A A064792 Reinhard Zumkeller (reinhard.zumkeller@lhsystems.com), Oct 17 2001
%I A030670
%S A030670 23,31,53,71,113,131,173,191,233,293,311,373,419,431,479,5323,593,
%T A030670 613,673,719,733,797,839,8923,971,1013,1031,10711,1091,11311,1277,
%U A030670 1319,1373,1399,1493,1511,1571,1637,16729,1733,17911,1811,1913,1931
%N A030670 Smallest prime whose decimal expansion begins (nontrivially) with the n-th prime.
%C A030670 Add more digits to p until reach another prime.
%t A030670 f[n_] := (k = 1; While[a = ToExpression[ ToString[n] <> ToString[k]]; ! PrimeQ[a], k++ ]; a); Table[ f[ Prime[n]], {n, 1, 45} ]
%Y A030670 See A064792 for another version. Cf. A065112.
%K A030670 nonn,base
%O A030670 1,1
%A A030670 Patrick De Geest (pdg@worldofnumbers.com)
%I A030680
%S A030680 23,31,53,71,113,1013,1319,1511,1811,1913,3137,3533,3733,3833,72719,
%T A030680 7573,7873,79757,9199,9293,1030111,105019,106013,113111,114113,
%U A030680 124213,127217,128213,133319,138311,139313,143413,147419,1545121
%N A030680 Smallest nontrivial extension of n-th palindromic prime which is a prime.
%K A030680 nonn,base
%O A030680 1,1
%A A030680 Patrick De Geest (pdg@worldofnumbers.com)
%I A006203 M5131
%S A006203 23,31,59,83,107,139,211,283,307,331,379,499,547,643,883,907
%N A006203 Discriminants of imaginary quadratic fields with class number 3 (negated).
%C A006203 Also n such that Q(sqrt(-n)) has class number 3.
%D A006203 H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
%D A006203 J. M. Masley, Where are the number fields with small class number?, pp. 221-242 of Number Theory Carbondale 1979, Lect. Notes Math. 751 (1982).
%H A006203 E. W. Weisstein, Link to a section of The World of Mathematics.
%H A006203 Index entries for sequences related to quadratic fields
%o A006203 For PARI code see A005847.
%Y A006203 Cf. A013658, A014602, A014603, A046002, ..., A046020. Cf. also A003173, A005847, ...
%K A006203 fini,nonn,full,nice
%O A006203 1,1
%A A006203 njas
%I A052160
%S A052160 23,31,61,73,83,131,233,271,331,353,383,433,443,503,541,571,677,751,
%T A052160 991,1013,1033,1063,1231,1283,1291,1321,1433,1453,1493,1543,1553,1601,
%U A052160 1613,1621,1657,1777,1861,1973,1987,2011,2063,2131,2207,2333,2341,2351
%N A052160 Isolated prime difference equals 6: d(n)=p(n+1)-p(n)=6 but d(n+1) and d(n-1) different from 6.
%C A052160 Compare to A047948 and A033451 which initial primes of X66Y and X666Y consecutive prime difference patterns, terms of A001223. No other "islands of 6" occur in A001223: X6Y,X66Y or X666Y.
%e A052160 Consecutive primes 17,19,23,29,31 gives 2,4,6,2,.. difference patterne in which the neighboring differences of 6 are not equal to 6. Remark that terms a(n)-6 can be prime but not immediate precedent one, like 23-6=17, but prior to 19 comes before 23.
%Y A052160 A001223, A033451, A047948.
%K A052160 nonn
%O A052160 1,1
%A A052160 Labos E. (labos@ana1.sote.hu), Jan 25 2000
%I A039410
%S A039410 23,32,123,132,203,213,230,231,234,235,236,237,238,239,243,253,263,273,
%T A039410 283,293,302,312,320,321,324,325,326,327,328,329,342,352,362,372,382,
%U A039410 392,423,432,523,532,623,632,723,732,823,832,923,932,1023,1032
%N A039410 Representation in base 10 has same nonzero number of 2's and 3's.
%K A039410 nonn,base,easy
%O A039410 0,1
%A A039410 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043233
%S A043233 23,32,123,132,223,230,320,332,423,432,523,532,623,632,723,732,
%T A043233 823,832,923,932,1023,1032,1123,1132,1223,1230,1320,1332,1423,
%U A043233 1432,1523,1532,1623,1632,1723,1732,1823,1832,1923,1932,2023
%N A043233 2 and 3 occur juxtaposed in the base 10 representation of n but not of n-1.
%K A043233 nonn,base
%O A043233 1,1
%A A043233 Clark Kimberling, ck6@cedar.evansville.edu
%I A044013
%S A044013 23,32,123,132,223,239,329,332,423,432,523,532,623,632,723,732,
%T A044013 823,832,923,932,1023,1032,1123,1132,1223,1239,1329,1332,1423,
%U A044013 1432,1523,1532,1623,1632,1723,1732,1823,1832,1923,1932,2023
%N A044013 2 and 3 occur juxtaposed in the base 10 representation of n but not of n+1.
%K A044013 nonn,base
%O A044013 1,1
%A A044013 Clark Kimberling, ck6@cedar.evansville.edu
%I A038356
%S A038356 23,33,59,69,95,105,113,119,125,137,138,139,140,142,167,177,183,189,
%T A038356 195,198,199,200,202,207,239,249,275,285,311,321,329,335,341,353,354,
%U A038356 355,356,358,383,393,399,405,411,414,415,416,418,423,455,465,491,501
%N A038356 Representation in base 6 has same nonzero number of 3's and 5's.
%K A038356 nonn,base,easy
%O A038356 0,1
%A A038356 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043129
%S A043129 23,33,59,69,95,105,131,138,167,177,198,213,239,249,275,285,311,
%T A043129 321,347,354,383,393,414,429,455,465,491,501,527,537,563,570,
%U A043129 599,609,630,645,671,681,707,717,743,753,779,786,815,825,828
%N A043129 3 and 5 occur juxtaposed in the base 6 representation of n but not of n-1.
%K A043129 nonn,base
%O A043129 1,1
%A A043129 Clark Kimberling, ck6@cedar.evansville.edu
%I A043909
%S A043909 23,33,59,69,95,105,131,143,167,177,203,213,239,249,275,285,311,
%T A043909 321,347,359,383,393,419,429,455,465,491,501,527,537,563,575,
%U A043909 599,609,635,645,671,681,707,717,743,753,779,791,815,825,863
%N A043909 3 and 5 occur juxtaposed in the base 6 representation of n but not of n+1.
%K A043909 nonn,base
%O A043909 1,1
%A A043909 Clark Kimberling, ck6@cedar.evansville.edu
%I A049851
%S A049851 23,34,35,56,57,78,79,710,711,1112,1113,1314,1315,1316,1317,1718,
%T A049851 1719,1920,1921,1922,1923,2324,2325,2326,2327,2328,2329,2930,2931,
%U A049851 3132,3133,3134,3135,3136,3137,3738,3739,3740,3741,4142,4143,4344
%N A049851 Concatenate "prevprime(n)" and "n".
%K A049851 nonn,base
%O A049851 3,1
%A A049851 njas
%I A045533
%S A045533 23,35,57,711,1113,1317,1719,1923,2329,2931,3137,3741,4143,
%T A045533 4347,4753,5359,5961,6167,6771,7173,7379,7983,8389,8997,
%U A045533 97101,101103,103107,107109,109113,113127,127131,131137
%N A045533 Concatenate n-th and (n+1)st primes.
%K A045533 nonn,base
%O A045533 0,1
%A A045533 njas
%I A059798
%S A059798 23,37,43,59,67,73,89,1237,1567,1787,1907,2141,2251,2297,2633,2963,
%T A059798 3019,4079,4253,4363,4583,6637,6701,6857,6967,7211,7321,7541,8147,8419,
%U A059798 8923,8969,9103,9323,9433,29303,29803,40093,40193,40493,40693,40993
%N A059798 Primes p such that |p - q| is a square, where q is the reversal of p.
%F A059798 Subtract reversal of prime from prime. Take absolute value. If a square, add to sequence.
%e A059798 a(7)=1237. Reverse is 7321 so 7321-1237=6084, and 6084 is a square whose root is 78.
%Y A059798 Cf. A059799.
%K A059798 easy,nonn,base
%O A059798 0,1
%A A059798 Enoch Haga (EnochHaga@msn.com), Feb 23 2001
%I A055114
%S A055114 23,37,53,59,61,83,103,107,113,127,137,139,149,151,197,211,223,227,229,
%T A055114 331,347,349,353,359,383,421,439,461,479,491,509,523,529,541,557,563,
%U A055114 569,607,631,739,751,757,761,769,797,809,811,821,823,827,829,839,851
%N A055114 Continued fraction for k/n contains a term >= 3 for every 1<=k= 3.
%K A055114 nonn
%O A055114 1,1
%A A055114 David Wilson (wilson@aprisma.com), Jun 16 2000
%I A063643
%S A063643 23,37,53,67,89,113,131,157,211,251,293,307,337,379,409,449,487,491,
%T A063643 499,503,631,683,701,719,751,769,787,919,941,953,991,1009,1039,1117,
%U A063643 1193,1201,1259,1381,1399,1439,1459,1471,1499,1511,1567,1709,1733,1759
%N A063643 Primes with 2 representations: p*q - 2 = u*v + 2 where p, q, u and v are prime.
%e A063643 A063643(25) = 751: 751 = A063637(60)= 753 - 2 = 3*251 - 2, 751 = A063638(55)= 749 + 2 = 7*107 + 2.
%Y A063643 A063637, A063638, A001358.
%K A063643 nonn
%O A063643 1,1
%A A063643 Reinhard Zumkeller (reinhard.zumkeller@lhsystems.com), Jul 21 2001
%I A057876
%S A057876 23,37,53,73,113,131,137,151,173,179,197,211,311,317,431,617,719,1531,
%T A057876 1831,1997,2113,2131,2237,2273,2297,2311,2797,3137,3371,4337,4373,4733,
%U A057876 4919,6173,7297,7331,7573,7873,8191,8311,8831,8837,12239,16673,19531
%N A057876 Remains prime (no leading zeros allowed) after deleting all occurrences of its digits d.
%e A057876 A larger example 1210778071 gives primes 12177871, 2077807, 110778071, 1210801 and 121077071 after dropping digits 0, 1, 2, 7 and 8.
%Y A057876 Cf. A057877-A057883, A051362, A034302-A034305.
%K A057876 nonn,base
%O A057876 0,1
%A A057876 Patrick De Geest (pdg@worldofnumbers.com), Oct 2000.
%I A051362
%S A051362 23,37,53,73,113,131,137,173,179,197,311,317,431,617,719,1013,1031,
%T A051362 1097,1499,1997,2239,2293,3137,4019,4919,6173,7019,7433,9677,10193,
%U A051362 10613,11093,19973,23833,26833,30011,37019,40013,47933,73331,74177
%N A051362 Remains prime if any digit is deleted (zeros allowed).
%Y A051362 Cf. A034302.
%K A051362 nonn,base,nice
%O A051362 0,1
%A A051362 Harvey P. Dale (hpd1@is2.nyu.edu), May 31 2000
%I A034302
%S A034302 23,37,53,73,113,131,137,173,179,197,311,317,431,617,719,1499,1997,
%T A034302 2239,2293,3137,4919,6173,7433,9677,19973,23833,26833,47933,73331,
%U A034302 74177,91733,93491,94397,111731,166931,333911,355933,477797,477977
%N A034302 Remains prime if any digit is deleted (zeros not allowed).
%Y A034302 Cf. A034303, A034304, A034305, A051362.
%K A034302 base,nonn,nice
%O A034302 0,1
%A A034302 dww
%I A057878
%S A057878 23,37,53,73,113,131,151,211,311,11111111111111111131,
%T A057878 11111111111111117111,11111111111131111111,11111111131111111111,
%U A057878 111111111111111111111113,111111111111111112111111
%N A057878 Primes with 2 distinct digits that remain prime (no leading zeros allowed) after deleting all occurrences of its digits d.
%Y A057878 Cf. A057876-A057883, A004022, A004023, A051362, A034302-A034305.
%K A057878 nonn,base
%O A057878 0,1
%A A057878 Patrick De Geest (pdg@worldofnumbers.com), Oct 2000.
%I A066629
%S A066629 23,37,53,73,113,137
%N A066629 Primes which are a nontrivial concatenation of primes.
%e A066629 113 is member as 11 and 3 are prime.
%K A066629 easy,more,nonn,base
%O A066629 1,1
%A A066629 Amarnath Murthy (amarnath_murthy@yahoo.com), Dec 28 2001
%I A019549
%S A019549 23,37,53,73,113,137,173,193,197,211,223,227,229,233,241,257,271,277,283,
%T A019549 293,311,313,317,331,337,347,353,359,367,373,379,383,389,397,433,523,541,
%U A019549 547,557,571,577,593,613,617,673,677,719,727,733,743,757,761,773,797,977
%N A019549 Primes formed by concatenating other primes.
%D A019549 Sylvester Smith, "A Set of Conjectures on Smarandache Sequences", Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.
%H A019549 M. L. Perez et al., eds., Smarandache Notions Journal
%K A019549 nonn,base
%O A019549 0,1
%A A019549 R. Muller (aol.com!Research37)
%I A066064
%S A066064 23,37,53,73,113,137,173,193,233,293,313,373,4111,433,4723,5323,593,
%T A066064 613,673,7129,733,797,8311,8923,977,1013,1033,10711,1093,11311,1277,
%U A066064 13147,1373,13913,1493,15131,15731,1637,16729,1733,17911,18119,1913
%N A066064 a(n) = pq in decimal notation where p = prime(n) and q is the smallest prime (A066065(n)) such that p.q is a prime.
%e A066064 A000040(2) = 3 and as 32, 33, and 35 are composite, the next prime 7 = A066065(2) yields a(2) = 37.
%K A066064 base,nonn
%O A066064 1,1
%A A066064 Reinhard Zumkeller (reinhard.zumkeller@lhsystems.com), Dec 01 2001
%I A050657
%S A050657 23,41,47,53,71,79,83,91,99,113,149,181,217,229,289,293,311,349,359,
%T A050657 361,379,417,421,433,517,523,541,587,593,617,619,661,669,699,701,719,
%U A050657 727,769,787,789,797,881,923,933,959,969,971,1057,1077,1149,1169,1327
%N A050657 n such that number of primes produced according to rules stipulated in Honaker's A048853 is 6.
%e A050657 Altering a(8)=91 gives 6 primes: 11, 31, 41, 61, 71 and 97.
%Y A050657 Cf. A048853, A050662, A050668.
%K A050657 nonn,base
%O A050657 1,1
%A A050657 Patrick De Geest (pdg@worldofnumbers.com), Jul 1999.
%I A050668
%S A050668 23,41,47,53,71,79,83,113,149,181,229,293,311,349,359,379,421,433,523,
%T A050668 541,587,593,617,619,661,701,719,727,769,787,797,881,971,1327,1373,
%U A050668 1399,1471,1543,1741,1747,1811,1889,1913,1979,2027,2129,2137,2161,2251
%N A050668 Primes p such that number of primes produced according to rules stipulated in Honaker's A048853 is 6.
%e A050668 Altering a(1)=23 gives 6 primes: 13, 43, 53, 73, 83 and 29.
%Y A050668 Cf. A048853, A050673, A050657.
%K A050668 nonn,base
%O A050668 1,1
%A A050668 Patrick De Geest (pdg@worldofnumbers.com), Jul 1999.
%I A037137
%S A037137 23,43,51,56,69,75,77,87,93,99,107,111,123,147,159,177,183,205,219,223,
%T A037137 227,231,235,253,267,271,285,291,295,299,301,303,317,327,349,357,363,
%U A037137 379,391,399,405,411,447,457,465,467,471,483,487,489,511,519,537,543
%N A037137 n such that A037134(n) = A037135(n) > 0.
%H A037137 Naohiro Nomoto, Sequence of Yin and Yang
%K A037137 nonn
%O A037137 0,1
%A A037137 Naohiro Nomoto (6284968128@geocities.co.jp)
%I A023264
%S A023264 23,43,83,109,193,379,389,569,643,659,853,1063,1129,1283,1423,1493,1759,
%T A023264 1789,1889,2003,2129,2293,2459,2713,2729,2879,2969,3373,3823,4519,4603,
%U A023264 4649,4663,4703,4783,4789,5023,5153,5209,5639,5653,5669,5693,5783,6203
%N A023264 Remains prime through 2 iterations of function f(x) = 8x + 9.
%K A023264 nonn
%O A023264 1,1
%A A023264 dww
%I A003859
%S A003859 1,23,45,45,231,231,252,253,483,770,770,990,990,1035,1035,1035,
%T A003859 1265,1771,2024,2277,3312,3520,5313,5544,5796,10395
%N A003859 Degrees of irreducible representations of group M24.
%D A003859 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%o A003859 (GAP) Display(CharacterTable("M24"));
%K A003859 nonn,fini,full
%O A003859 1,2
%A A003859 njas
%I A058545
%S A058545 23,45,57,91,161,291,481,911,1821,3031,5621,9759,16261,29199,48661,
%T A058545 97321,180731,361461,595359,914661,1302543,1810989,2732295,3594645,
%U A058545 4393215,5513805,5482215,6700245,8913663,12588669,19116087,31171125
%N A058545 Trajectory of 23 under map that sends x to 3x - sigma(x), where sigma(x) is the sum of the divisors of x.
%C A058545 Is it a theorem that this sequence goes to infinity?
%Y A058545 Cf. A037159, A058541, A058542.
%K A058545 nonn
%O A058545 0,1
%A A058545 njas, Dec 24 2000
%I A030656
%S A030656 1,23,45,67,89,1011,1213,1415,1617,1819,2021,2223,2425,2627,2829,3031,
%T A030656 3233,3435,3637,3839,4041,4243,4445,4647,4849,5051,5253,5455,5657,5859,
%U A030656 6061,6263,6465,6667,6869,7071,7273,7475,7677,7879,8081,8283,8485,8687
%N A030656 Pair up the numbers.
%Y A030656 Cf. A030655.
%K A030656 base,nonn,easy
%O A030656 0,2
%A A030656 abdelkader.maghraoui (maghraoui.faure.recherche.entpe@obelix.entpe.fr)
%E A030656 More terms from Erich Friedman (erich.friedman@stetson.edu).
%I A040506
%S A040506 23,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,
%T A040506 46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,
%U A040506 46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,46
%N A040506 Continued fraction for sqrt(530).
%H A040506 Index entries for continued fractions for constants
%p A040506 with(numtheory): Digits:=300: convert(evalf(sqrt(530)),confrac);
%K A040506 nonn,cofr,easy
%O A040506 0,1
%A A040506 njas
%I A025022
%S A025022 23,46,47,73,94,97,146,167,193,194,263,313,334,337,383,386,433,457,503,
%T A025022 526,529,577,626,647,673,674,743,766,863,866,887,914,937,983,1006,1033,
%U A025022 1058,1081,1103,1153,1154,1223,1294,1297,1346,1367,1486,1487,1583,1607
%N A025022 Least non-residue mod n is 5.
%K A025022 nonn
%O A025022 1,1
%A A025022 dww
%I A048845
%S A048845 1,23,46,69,92,115,138,161,184,207,230,253,276,299,322,345,368,
%T A048845 391,414,437,460,483,506,529,529,552,575,598,621,644,667,690,713,
%U A048845 736,759,782,805,828,851,874,897,920,943,966,989,1012,1035,1058
%N A048845 Least positive integer k for which 23^n divides k!.
%Y A048845 See A007843 for more information.
%K A048845 nonn
%O A048845 0,2
%A A048845 Charles T. Le (charlestle@yahoo.com)
%I A008605
%S A008605 0,23,46,69,92,115,138,161,184,207,230,253,276,299,322,
%T A008605 345,368,391,414,437,460,483,506,529,552,575,598,621,
%U A008605 644,667,690,713,736,759,782,805,828,851,874,897,920
%N A008605 Multiples of 23.
%H A008605 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 335
%K A008605 nonn
%O A008605 0,2
%A A008605 njas
%I A038152
%S A038152 23,46,69,92,115,138,161,185,208,231,254,277,300,232,347,370,393,416,
%T A038152 439,462,485,509,532,555,578,601,624,647,671
%N A038152 Beatty sequence for e^pi.
%H A038152 Index entries for sequences related to Beatty sequences
%F A038152 a(n)=int(n*23.1406926327...)
%K A038152 nonn
%O A038152 1,1
%A A038152 Felice Russo (felice.russo@katamail.com)
%I A052159
%S A052159 23,47,53,83,131,167,173,233,251,257,263,353,383,443,503,557,563,587,
%T A052159 593,647,653,677,941,947,971,977,1013,1097,1103,1181,1187,1217,1223,
%U A052159 1283,1361,1367,1433,1493,1553,1601,1613,1901,1907,1973,2063,2207,2333
%N A052159 Lower prime of a difference of 6 (G-minor-6 primes) between consecutive primes of 6k+5 form.
%C A052159 The corresponding larger primes (G-major-6 primes) have also 6k+5 form.
%F A052159 A031924[n]==5 (mod 6)
%e A052159 a(1)=23 since a(1)+6=29 is the next prime and 29=6*4+5
%Y A052159 A031924, A031925.
%K A052159 nonn
%O A052159 1,1
%A A052159 Labos E. (labos@ana1.sote.hu), Jan 25 2000
%I A042046
%S A042046 23,47,70,117,304,421,1146,1567,12115,183292,378699,2455486,
%T A042046 12656129,293546453,1480388394,9175876817,19832142028,306658007237,
%U A042046 2166438192687,2473096199924,7112630592535,9585726792459
%N A042046 Numerators of continued fraction convergents to sqrt(547).
%Y A042046 Cf. A042047.
%K A042046 nonn,cofr,easy
%O A042046 0,1
%A A042046 njas
%I A042044
%S A042044 23,47,70,187,257,701,32503,65707,98210,262127,360337,982801,
%T A042044 45569183,92121167,137690350,367501867,505192217,1377886301,
%U A042044 63887962063,129153810427,193041772490,515237355407,708279127897
%N A042044 Numerators of continued fraction convergents to sqrt(546).
%Y A042044 Cf. A042045.
%K A042044 nonn,cofr,easy
%O A042044 0,1
%A A042044 njas
%I A042042
%S A042042 23,47,70,607,677,1961,90883,183727,274610,2380607,2655217,
%T A042042 7691041,356443103,720577247,1077020350,9336740047,10413760397,
%U A042042 30164260841,1397969759083,2826103779007,4224073538090
%N A042042 Numerators of continued fraction convergents to sqrt(545).
%Y A042042 Cf. A042043.
%K A042042 nonn,cofr,easy
%O A042042 0,1
%A A042042 njas
%I A001124 M5132 N2224
%S A001124 23,47,73,97,103,157,167,193,263,277,307,383,397,433,503,577,647,673,
%T A001124 683,727,743,863,887,937,967,983,1033,1093,1103,1153,1163,1223,1367,
%U A001124 1487,1543,1583,1607,1777,1823,1847,1933
%N A001124 Primes with 5 as smallest primitive root.
%D A001124 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 A001124 M. Kraitchik, Recherches sur la Th\'{e}orie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 57.
%H A001124 Index entries for primes by primitive root
%t A001124 << NumberTheory`NumberTheoryFunctions`; Prime[ Select[ Range[200], PrimitiveRoot[ Prime[ # ] ] == 5 & ] ]
%Y A001124 Cf. A001122, A001123, A001125, etc.
%K A001124 nonn
%O A001124 1,1
%A A001124 njas
%E A001124 More terms from Robert G. Wilson v (rgwv@kspaint.com), May 10 2001
%I A054821
%S A054821 23,47,89,113,233,293,317,353,359,389,409,449,467,509,577,647,683,691,
%T A054821 839,863,887,919,1039,1069,1097,1201,1237,1283,1307,1327,1381,1433,
%U A054821 1439,1459,1493,1499,1559,1613,1627,1637,1709,1889,2003,2039,2099,2179
%N A054821 Third term of weak prime quartet: p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m).
%Y A054821 Cf. A051635, A054800-A054840.
%K A054821 nonn
%O A054821 0,1
%A A054821 Henry Bottomley (se16@btinternet.com), Apr 10 2000
%I A039374
%S A039374 23,47,104,128,167,176,194,203,207,208,210,211,213,214,215,221,230,239,
%T A039374 266,290,347,371,407,416,423,424,426,427,429,430,431,434,443,461,470,
%U A039374 479,509,533,590,614,671,695,752,776,833,857,896,905,923,932,936
%N A039374 Representation in base 9 has same nonzero number of 2's and 5's.
%K A039374 nonn,base,easy
%O A039374 0,1
%A A039374 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043197
%S A043197 23,47,104,128,185,207,266,290,347,371,423,452,509,533,590,614,
%T A043197 671,695,752,776,833,857,914,936,995,1019,1076,1100,1152,1181,
%U A043197 1238,1262,1319,1343,1400,1424,1481,1505,1562,1586,1643,1665
%N A043197 2 and 5 occur juxtaposed in the base 9 representation of n but not of n-1.
%K A043197 nonn,base
%O A043197 1,1
%A A043197 Clark Kimberling, ck6@cedar.evansville.edu
%I A043977
%S A043977 23,47,104,128,185,215,266,290,347,371,431,452,509,533,590,614,
%T A043977 671,695,752,776,833,857,914,944,995,1019,1076,1100,1160,1181,
%U A043977 1238,1262,1319,1343,1400,1424,1481,1505,1562,1586,1643,1673
%N A043977 2 and 5 occur juxtaposed in the base 9 representation of n but not of n+1.
%K A043977 nonn,base
%O A043977 1,1
%A A043977 Clark Kimberling, ck6@cedar.evansville.edu
%I A042048
%S A042048 23,47,117,398,515,2973,3488,37853,41341,244558,285899,
%T A042048 1102255,2490409,6083073,282311767,570706607,1423724981,
%U A042048 4841881550,6265606531,36169914205,42435520736,460525121565
%N A042048 Numerators of continued fraction convergents to sqrt(548).
%Y A042048 Cf. A042049.
%K A042048 nonn,cofr,easy
%O A042048 0,1
%A A042048 njas
%I A042050
%S A042050 23,47,164,1523,16917,18440,35357,53797,89154,410413,499567,
%T A042050 909980,1409547,2319527,26924344,244638623,760840213,1766319049,
%U A042050 82011516467,165789351983,579379572416,5380205503727,59761640113413
%N A042050 Numerators of continued fraction convergents to sqrt(549).
%Y A042050 Cf. A042051.
%K A042050 nonn,cofr,easy
%O A042050 0,1
%A A042050 njas
%I A065017
%S A065017 23,47,167,359,1847,3719,10607,19319,97967,177239,273527,657719,
%T A065017 1042439,1104599,1329407,1515359,1745039,2042039,4464767,5013119,
%U A065017 5148359,9740639,11095559,11377127,12538679,16024007,16410599,16752647
%N A065017 p*q + p + q is prime, where (p, q=p+2) are twin primes.
%C A065017 The resulting prime can never be a twin prime since the odd number preceding it is divisible by three and the following odd number is a perfect square.
%F A065017 p^2+4*p+2.
%e A065017 (3*5) + (3+5) = 23
%t A065017 NextPrim[n_] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; Return[k]]; k = 1; Do[k = NextPrim[k]; If[ PrimeQ[k + 2], p = k*(k + 2) + 2k + 2; If[ PrimeQ[p], Print[p]]], {n, 1, 700} ]
%Y A065017 A049001, A049002
%K A065017 nonn
%O A065017 0,1
%A A065017 Stephan Wagler (stephanwagler@aol.com), Nov 01 2001
%I A042052
%S A042052 23,47,211,258,727,985,1712,12969,53588,388085,441673,829758,
%T A042052 2101189,2930947,13824977,30580901,1420546423,2871673747,
%U A042052 12907241411,15778915158,44465071727,60243986885,104709058612
%N A042052 Numerators of continued fraction convergents to sqrt(550).
%Y A042052 Cf. A042053.
%K A042052 nonn,cofr,easy
%O A042052 0,1
%A A042052 njas
%I A054693
%S A054693 23,47,241,241,523,677,677,677,2861,10733,10733,13421,13421,13421,
%T A054693 13421,13421,13421,13421,13421,13421,13421,13421,13421,61631,61631,
%U A054693 61631,61631,61631,61631,333793,333793,333793,333793,333793,333793
%N A054693 n consecutive primes differ by 6 or more starting at a(n).
%K A054693 nonn
%O A054693 2,1
%A A054693 Jeff Burch (gburch@erols.com), Apr 19 2000
%E A054693 More terms from Larry Reeves (larryr@acm.org), Nov 09 2000
%I A052229
%S A052229 23,47,251,167,727,433,941,1187,1453,1367,2417,4597,2207,3761,4657,
%T A052229 4451,5557,6317,7517,8923,9043,17707,15227,12823,10607,33487,28663,
%U A052229 29717,50417,31567,24793,24043,28753,28837,29983,29173,59951,45497
%N A052229 a(n) is first prime p from A031924 such that A052180(p) = Prime[n].
%Y A052229 Cf. A031924, A031925, A052158, A052180.
%K A052229 nonn
%O A052229 3,1
%A A052229 Labos E. (labos@ana1.sote.hu), Feb 01 2000
%I A054679
%S A054679 23,47,251,1741,1741,19471,118801,148531,148531,406951,1820111,2339041
%N A054679 n consecutive primes differ by a multiple of 6 starting at a(n).
%Y A054679 Cf. A054678, A054680.
%K A054679 nonn,more
%O A054679 2,1
%A A054679 Jeff Burch (gburch@erols.com), Apr 18 2000
%E A054679 More terms from Larry Reeves (larryr@acm.org), Nov 09 2000
%I A054203
%S A054203 23,47,251,1889,1741,19471,118801,148537,148531,406951
%N A054203 Smallest primes followed by n consecutive (unequal) prime differences, each divided by 6.
%C A054203 This is a "modular arithmetic progression" of successive primes, modulo 6.
%H A054203 Index entries for sequences related to primes in arithmetic progressions
%e A054203 n=1 and 23 is followed by d=6 to give 29, a prime; n=5 a(5)=1741 is followed by {d}={6,6,6,18,6} and results in {1741,1747,1753,1759,1777,1783} consecutive prime sequence, while a(10)=406951 prime is followed by {18,12,12,30,24,12,24,36,18,12} consecutive d-pattern.
%Y A054203 A033451, A052243, A052239.
%K A054203 more,nonn
%O A054203 1,1
%A A054203 Labos E. (labos@ana1.sote.hu), May 17 2000
%I A042054
%S A042054 23,47,399,446,3967,8380,389447,787274,6687639,7474913,
%T A042054 66486943,140448799,6527131697,13194712193,112084829241,
%U A042054 125279541434,1114321160713,2353921862860,109394726852273
%N A042054 Numerators of continued fraction convergents to sqrt(551).
%Y A042054 Cf. A042055.
%K A042054 nonn,cofr,easy
%O A042054 0,1
%A A042054 njas
%I A042056
%S A042056 23,47,2185,4417,205367,415151,19302313,39019777,1814212055,
%T A042056 3667443887,170516630857,344700705601,16026749088503,32398198882607,
%U A042056 1506343897688425,3045085994259457,141580299633623447,286205685261506351
%N A042056 Numerators of continued fraction convergents to sqrt(552).
%Y A042056 Cf. A042057.
%K A042056 nonn,cofr,easy
%O A042056 0,1
%A A042056 njas
%I A044100
%S A044100 23,48,73,98,115,123,148,173,198,223,240,248,273,298,323,348,
%T A044100 365,373,398,423,448,473,490,498,523,548,573,575,615,623,648,
%U A044100 673,698,723,740,748,773,798,823,848,865,873,898,923,948,973
%N A044100 String 4,3 occurs in the base 5 representation of n but not of n-1.
%K A044100 nonn,base
%O A044100 1,1
%A A044100 Clark Kimberling, ck6@cedar.evansville.edu
%I A044481
%S A044481 23,48,73,98,119,123,148,173,198,223,244,248,273,298,323,348,369,373,
%T A044481 398,423,448,473,494,498,523,548,573,599,619,623,648,673,698,723,744,
%U A044481 748,773,798,823,848,869,873,898,923,948,973
%N A044481 String 4,3 occurs in the base 5 representation of n but not of n+1.
%K A044481 nonn,base
%O A044481 1,1
%A A044481 Clark Kimberling, ck6@cedar.evansville.edu
%I A029493
%S A029493 1,23,49,73,79,91,97,139,349,511,529,553,577,20401,25189,86227,92989
%N A029493 n divides the (left) concatenation of all numbers <= n written in base 24 (most significant digit on left).
%C A029493 The next term is > 410000. - Larry Reeves, Jan 16, 2002
%Y A029493 Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.
%K A029493 nonn,base,more
%O A029493 1,2
%A A029493 Olivier Gerard (ogerard@ext.jussieu.fr)
%E A029493 More terms from Larry Reeves (larryr@acm.org), May 24 2001
%I A063321
%S A063321 1,23,49,77,103,131,157,185,211,239,265,293,319,347,373,401,427,
%T A063321 455,481,509,535,563,589,617,643,671,697,725,751,779,805,833,859,
%U A063321 887,913,941,967,995,1021,1049,1075,1103,1129,1157,1183,1211,1237
%V A063321 -1,23,49,77,103,131,157,185,211,239,265,293,319,347,373,401,427,
%W A063321 455,481,509,535,563,589,617,643,671,697,725,751,779,805,833,859,
%X A063321 887,913,941,967,995,1021,1049,1075,1103,1129,1157,1183,1211,1237
%N A063321 Dimension of the space of weight n cuspidal newforms for Gamma_1( 48 ).
%H A063321 William A. Stein (was@math.berkeley.edu), Dimensions of the spaces S_k^{new}(Gamma_1(N))
%H A063321 William A. Stein (was@math.berkeley.edu), The modular forms database
%K A063321 sign,done
%O A063321 2,2
%A A063321 njas, Jul 14 2001
%I A055782
%S A055782 23,53,73,113,173,193,233,293,313,373,433,593,613,673,733,1013,1033,
%T A055782 1093,1373,1493,1733,1913,1933,1973,1993,2113,2273,2293,2333,2393,2633,
%U A055782 2693,2713,2833,3313,3373,3533,3593,3673,3733,3793,3833,4013,4093,4493
%N A055782 Primes q of form q=10p+3, where p is also prime.
%e A055782 5413=541*10+3, 3 glued to 541
%Y A055782 Cf. A005384, A005385. Apart from first term, same as A057667.
%K A055782 nonn,easy
%O A055782 1,1
%A A055782 Labos E. (labos@ana1.sote.hu), Jul 13 2000
%I A051650
%S A051650 0,23,53,120,211,1340,1341,1342,1343,1344,2179,3967,15704,15705,16033,
%T A051650 19634,19635,24281,31428,31429,31430,31431,31432,31433,38501,58831,
%U A051650 155964,203713,206699,370310,370311,370312,370313,370314,370315,370316
%N A051650 Lonely numbers: distance to closest prime sets a new record.
%e A051650 23 is 4 units away from the closest prime (not including itself), so 23 is in the sequence.
%Y A051650 Related sequences: A023186-A023188, A046929-A046931, A051650, A051652, A051697-A051702, A051728-A051730.
%Y A051650 Distances are in A051730.
%K A051650 nonn,easy,nice
%O A051650 0,2
%A A051650 njas
%E A051650 More terms from James A. Sellers (sellersj@math.psu.edu), Dec 23 1999, and from Jud McCranie (jud.mccranie@mindspring.com), Jun 16 2000
%I A049438
%S A049438 23,53,131,173,233,263,563,593,653,1013,1223,1283,1601,1613,2333,2543,
%T A049438 2963,3323,3533,3761,3911,3923,4013,4211,4253,4643,4793,5003,5273,5471,
%U A049438 5843,5861,6263,6353,6563,6653,6863,7121,7451,7481,7541,7583
%N A049438 p, p+6 and p+8 are all primes (A023202) but p+2 is not.
%Y A049438 Cf. A007530, A023202, A031926, A046134, A046138, A049436, A049437.
%K A049438 nonn
%O A049438 1,1
%A A049438 Labos E. (labos@ana1.sote.hu)
%I A045345
%S A045345 1,23,53,853,11869,117267,339615,3600489,96643287
%N A045345 n divides sum of first n primes.
%H A045345 E. W. Weisstein, Link to a section of The World of Mathematics.
%K A045345 nonn,nice
%O A045345 1,2
%A A045345 Jud McCranie (jud.mccranie@mindspring.com)
%I A053236
%S A053236 23,54,59,84,114,138,149,172,177,232,257,281,293,311,355,392,417,422,
%T A053236 434,445,481,506,561,596,601,644,656,686,715,745,763,775,798,809,853,
%U A053236 864,944,955,979,984,1013,1018,1061,1072,1140,1164,1187,1192,1222,1227
%N A053236 n such that A053230(n) = 4.
%p A053236 with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..8000)];
%p A053236 seq( `if`(f[i+1] - f[i] = 4,i,print( )), i=1..3000);
%Y A053236 Cf. A000203, A053224, A053230, A053231, A053232, A053233, A053234, A053235, A053237.
%K A053236 nonn
%O A053236 1,1
%A A053236 Asher Auel (asher.auel@reed.edu) Jan 10, 2000
%I A033657
%S A033657 23,55,110,121,242,484,968,1837,9218,17347,91718,173437,
%T A033657 907808,1716517,8872688,17735476,85189247,159487405,664272356,
%U A033657 1317544822,3602001953,7193004016,13297007933,47267087164
%N A033657 Reverse and Add!.
%K A033657 nonn
%O A033657 1,1
%A A033657 njas
%I A007795
%S A007795 23,57,1113,1719,2329,3137,4143,4753,5961,6771,
%T A007795 7379,8389,97101,103107,109113,127131,137139,149151,
%U A007795 157163,167173,179181,191193,197199,211223,227229
%N A007795 Juxtapose pairs of primes.
%K A007795 nonn,base
%O A007795 1,1
%A A007795 bmoore@artemis.ess.ucla.edu (William B. Moore)
%I A039346
%S A039346 23,58,87,122,135,143,159,167,175,183,184,185,187,188,189,190,215,250,
%T A039346 279,314,343,378,407,442,450,458,464,465,467,468,469,470,474,482,490,
%U A039346 498,535,570,599,634,647,655,671,679,687,695,696,697,699,700,701,702
%N A039346 Representation in base 8 has same nonzero number of 2's and 7's.
%K A039346 nonn,base,easy
%O A039346 0,1
%A A039346 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A043169
%S A043169 23,58,87,122,151,184,215,250,279,314,343,378,407,442,464,506,
%T A043169 535,570,599,634,663,696,727,762,791,826,855,890,919,954,976,
%U A043169 1018,1047,1082,1111,1146,1175,1208,1239,1274,1303,1338,1367
%N A043169 2 and 7 occur juxtaposed in the base 8 representation of n but not of n-1.
%K A043169 nonn,base
%O A043169 1,1
%A A043169 Clark Kimberling, ck6@cedar.evansville.edu
%I A043949
%S A043949 23,58,87,122,151,191,215,250,279,314,343,378,407,442,471,506,
%T A043949 535,570,599,634,663,703,727,762,791,826,855,890,919,954,983,
%U A043949 1018,1047,1082,1111,1146,1175,1215,1239,1274,1303,1338,1367
%N A043949 2 and 7 occur juxtaposed in the base 8 representation of n but not of n+1.
%K A043949 nonn,base
%O A043949 1,1
%A A043949 Clark Kimberling, ck6@cedar.evansville.edu
%I A005111 M5133
%S A005111 23,59,67,83,89,107,173,199,227,233,263,311,317,331,349,353,367,373,383,
%T A005111 397,419,431,463,479,503,509,523,563,569,587,607,617,619,661,683,727,733
%N A005111 Class 3- primes.
%D A005111 R. K. Guy, Unsolved Problems in Number Theory, A18.
%K A005111 nonn
%O A005111 1,1
%A A005111 njas, Simon Plouffe (plouffe@math.uqam.ca)
%I A044125
%S A044125 23,59,95,131,138,167,203,239,275,311,347,354,383,419,455,491,
%T A044125 527,563,570,599,635,671,707,743,779,786,815,828,887,923,959,
%U A044125 995,1002,1031,1067,1103,1139,1175,1211,1218,1247,1283,1319,1355
%N A044125 String 3,5 occurs in the base 6 representation of n but not of n-1.
%K A044125 nonn,base
%O A044125 1,1
%A A044125 Clark Kimberling, ck6@cedar.evansville.edu
%I A044506
%S A044506 23,59,95,131,143,167,203,239,275,311,347,359,383,419,455,491,527,563,
%T A044506 575,599,635,671,707,743,779,791,815,863,887,923,959,995,1007,1031,
%U A044506 1067,1103,1139,1175,1211,1223,1247,1283,1319,1355
%N A044506 String 3,5 occurs in the base 6 representation of n but not of n+1.
%K A044506 nonn,base
%O A044506 1,1
%A A044506 Clark Kimberling, ck6@cedar.evansville.edu
%I A033217
%S A033217 23,59,101,167,173,211,223,271,307,317,347,449,463,593,
%T A033217 599,607,691,719,809,821,829,853,877,883,991,997,1097,1117,
%U A033217 1151,1163,1181,1231,1319,1451,1453,1481,1553,1613,1669
%N A033217 Primes of form x^2+23*y^2.
%C A033217 If x>0, then tau(p) = 2 mod 23 - comment from Jud McCranie (jud.mccranie@mindspring.com).
%D A033217 Lure of the Integers, Joe Roberts, "Integer 23 - the Tau function".
%D A033217 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.
%Y A033217 Cf. A000594.
%K A033217 nonn,nice
%O A033217 0,1
%A A033217 njas
%I A055821
%S A055821 1,23,60,122,217,354,543,795,1122,1537,2054,2688,3455,4372,5457,6729,
%T A055821 8208,9915,11872,14102,16629,19478,22675,26247,30222,34629,39498,44860,
%U A055821 50747,57192,64229,71893,80220,89247
%N A055821 T(n,n-4), array T as in A055818.
%K A055821 nonn
%O A055821 4,2
%A A055821 Clark Kimberling, ck6@cedar.evansville.edu, May 28 2000
%I A031342
%S A031342 23,61,103,151,197,251,307,359,419,463,523,593,643,701,761,827,883,
%T A031342 953,1019,1069,1129,1213,1279,1321,1427,1481,1543,1601,1663,1733,
%U A031342 1801,1877,1951,2017,2087,2143,2239,2297,2371,2423,2521,2593
%N A031342 (9n)-th prime.
%p A031342 ithprime(9*n)
%K A031342 nonn
%O A031342 1,1
%A A031342 Jeff Burch (jmburch@osprey.smcm.edu)
%I A001346 M5134
%S A001346 23,65,261,1370,8742,65304,557400,5343120,56775600,661933440,
%T A001346 8397406080,115123680000,1695705580800,26701944192000,447579574041600,
%U A001346 7955978033203200,149473718634240000
%N A001346 Sum (n+k)! C(4,k), k = 0 . . 4.
%D A001346 Biondi, E.; Divieti, L.; Guardabassi, G.; Counting paths, circuits, chains and cycles in graphs: A unified approach. Canad. J. Math. 22 1970 22-35.
%K A001346 nonn
%O A001346 -1,1
%A A001346 njas
%I A051875
%S A051875 0,1,23,66,130,215,321,448,596,765,955,1166,1398,1651,1925,2220,
%T A051875 2536,2873,3231,3610,4010,4431,4873,5336,5820,6325,6851,7398,7966,
%U A051875 8555,9165,9796,10448,11121,11815,12530,13266,14023,14801,15600
%N A051875 23-gonal numbers.
%D A051875 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
%F A051875 a(n)=n(21n-19)/2.
%K A051875 nonn
%O A051875 0,3
%A A051875 njas, Dec 15 1999
%I A059241
%S A059241 23,67,89,199,353,397,419,463,617,661,683,727,859,881,947,991,1123,
%T A059241 1277,1409,1453,1607,1783,1871,2003,2069,2113,2179,2267,2311,2333,2377,
%U A059241 2399,2531,2663,2707,2729,2861,2927,2971,3037,3169,3191,3257,3301,3323
%N A059241 Primes p such that x^11 = 2 has no solution mod p.
%C A059241 Complement of A049543 relative to A000040.
%Y A059241 Cf. A000040, A049543.
%K A059241 easy,nonn
%O A059241 0,1
%A A059241 Klaus Brockhaus (klaus-brockhaus@t-online.de), Jan 21 2001
%I A030458
%S A030458 23,67,89,1213,3637,4243,5051,5657,6263,6869,7879,8081,9091,9293,
%T A030458 9697,102103,108109,120121,126127,138139,150151,156157,180181,
%U A030458 186187,188189,192193,200201,216217,242243,246247,252253,270271
%N A030458 Primes formed by concatenating n with n+1.
%Y A030458 A052089, A052087, A052088.
%K A030458 nonn,base
%O A030458 0,1
%A A030458 Patrick De Geest (pdg@worldofnumbers.com)
%I A006055 M5135
%S A006055 23,67,89,4567,78901,678901,23456789,45678901,9012345678901,
%T A006055 789012345678901,56789012345678901234567890123,
%U A006055 90123456789012345678901234567
%N A006055 Primes with consecutive digits.
%D A006055 J. S. Madachy, Consecutive-digit primes - again, J. Rec. Math., 5 (No. 4, 1972), 253-254.
%D A006055 D. Zwillinger, Consecutive-Digit Primes - In Different Bases, J. Rec. Math., 10 (1972), 32-33.
%D A006055 R. C. Schroeppel, personal communication, 1991.
%Y A006055 A subset of A006510.
%K A006055 nonn
%O A006055 1,1
%A A006055 njas
%I A053559
%S A053559 23,67,89,131132133134135136137138139140141
%N A053559 Primes formed by concatenating n consecutive increasing numbers starting with a palindrome and ending with the next consecutive palindrome.
%Y A053559 Cf. A000040, A002113.
%K A053559 base,nonn
%O A053559 1,1
%A A053559 G. L. Honaker, Jr. (curios@bvub.com), Jan 16 2000
%I A031376
%S A031376 23,67,109,167,227,277,347,401,461,523,599,653,727,797,859,937,1009,
%T A031376 1063,1129,1217,1289,1367,1447,1499,1579,1637,1723,1801,1879,1979,
%U A031376 2039,2113,2207,2281,2351,2417,2521,2609,2683,2731,2803,2897,2971
%N A031376 Primes p(10n-1).
%K A031376 nonn
%O A031376 1,1
%A A031376 Jeff Burch (jmburch@osprey.smcm.edu)
%I A042036
%S A042036 23,70,93,163,582,745,1327,2072,46911,48983,95894,144877,
%T A042036 530525,675402,1205927,4293183,198692345,600370218,799062563,
%U A042036 1399432781,4997360906,6396793687,11394154593,17790948280
%N A042036 Numerators of continued fraction convergents to sqrt(542).
%Y A042036 Cf. A042037.
%K A042036 nonn,cofr,easy
%O A042036 0,1
%A A042036 njas
%I A042034
%S A042034 23,70,93,535,628,5559,11746,52543,64289,181121,607652,
%T A042034 788773,1396425,16149448,243638145,503425738,1250489621,
%U A042034 1753915359,3004404980,4758320339,7762725319,12521045658
%N A042034 Numerators of continued fraction convergents to sqrt(541).
%Y A042034 Cf. A042035.
%K A042034 nonn,cofr,easy
%O A042034 0,1
%A A042034 njas
%I A042038
%S A042038 23,70,233,769,1002,14797,15799,62194,202381,669337,30991883,
%T A042038 93644986,311926841,1029425509,1341352350,19808358409,21149710759,
%U A042038 83257490686,270922182817,896024039137,41488027983119,125360107988494
%N A042038 Numerators of continued fraction convergents to sqrt(543).
%Y A042038 Cf. A042039.
%K A042038 nonn,cofr,easy
%O A042038 0,1
%A A042038 njas
%I A042040
%S A042040 23,70,793,2449,113447,342790,3884137,11995201,555663383,
%T A042040 1678985350,19024502233,58752492049,2721639136487,8223669901510,
%U A042040 93182008053097,287769694060801,13330587934849943,40279533498610630
%N A042040 Numerators of continued fraction convergents to sqrt(544).
%Y A042040 Cf. A042041.
%K A042040 nonn,cofr,easy
%O A042040 0,1
%A A042040 njas
%I A035072
%S A035072 23,71,224,708,2237,7072,22361,70711,223607,707107,2236068,7071068,
%T A035072 22360680,70710679,223606798,707106782,2236067978,7071067812,
%U A035072 22360679775,70710678119,223606797750,707106781187,2236067977500
%N A035072 a(n) is root of square starting with digit 5: first term of runs.
%F A035072 ceiling(sqrt((5*10^n)), n > 1.
%Y A035072 Cf. A017936.
%K A035072 nonn
%O A035072 2,1
%A A035072 Patrick De Geest (pdg@worldofnumbers.com), Nov 1998.
%I A044161
%S A044161 23,72,121,161,170,219,268,317,366,415,464,504,513,562,611,660,
%T A044161 709,758,807,847,856,905,954,1003,1052,1101,1127,1190,1199,1248,
%U A044161 1297,1346,1395,1444,1493,1533,1542,1591,1640,1689,1738,1787
%N A044161 String 3,2 occurs in the base 7 representation of n but not of n-1.
%K A044161 nonn,base
%O A044161 1,1
%A A044161 Clark Kimberling, ck6@cedar.evansville.edu
%I A044542
%S A044542 23,72,121,167,170,219,268,317,366,415,464,510,513,562,611,660,709,758,
%T A044542 807,853,856,905,954,1003,1052,1101,1175,1196,1199,1248,1297,1346,1395,
%U A044542 1444,1493,1539,1542,1591,1640,1689,1738,1787
%N A044542 String 3,2 occurs in the base 7 representation of n but not of n+1.
%K A044542 nonn,base
%O A044542 1,1
%A A044542 Clark Kimberling, ck6@cedar.evansville.edu
%I A052073
%S A052073 23,83,113,1123,200003,328127,381289
%N A052073 Nextprime(p(n)) is substring of p(n)^2.
%Y A052073 Cf. A052074, A052075, A052076.
%K A052073 nonn,base,more,nice
%O A052073 0,1
%A A052073 Patrick De Geest (pdg@worldofnumbers.com), Jan 2000.
%I A060456
%S A060456 1,23,85,230,535,1124,2197,4071,7228,12391,20631,33506,53252,83047,
%T A060456 127358,192400,286751,422150,614551,885479,1263794,1787960,2508951,
%U A060456 3493969,4831163,6635623,9056943,12288743,16580630,22253151,29716459
%N A060456 Floor ( Exp(Pi*Sqrt(n)) ).
%C A060456 Some of these values are very close to integers, e.g. a(17)= 422150.9976756804516....
%H A060456 Exp(Pi*Sqrt(n)) Page
%p A060456 Digits:=100; for n from 0 to 40 do printf(`%d,`,floor( exp(Pi*sqrt(n)))) od:
%K A060456 easy,huge,nonn
%O A060456 0,2
%A A060456 Jason Earls (jcearls@kskc.net), Apr 08 2001
%E A060456 More terms from James A. Sellers (sellersj@math.psu.edu), Apr 11 2001
%I A056580
%S A056580 1,23,85,231,535,1124,2198,4072,7228,12392,20632,33506,53252,83048,
%T A056580 127359,192401,286751,422151,614552,885480,1263795,1787960,2508952,
%U A056580 3493970,4831164,6635624,9056943,12288744,16580631,22253151,29716459
%N A056580 round[e^(n*sqrt(pi))].
%Y A056580 Cf. A019296, A019297, A035484, A056581.
%K A056580 nonn
%O A056580 0,2
%A A056580 Henry Bottomley (se16@btinternet.com), Jun 30 2000
%I A010011
%S A010011 1,23,86,191,338,527,758,1031,1346,1703,2102,2543,3026,
%T A010011 3551,4118,4727,5378,6071,6806,7583,8402,9263,10166,11111,
%U A010011 12098,13127,14198,15311,16466,17663,18902,20183,21506
%N A010011 a(0)=1, a(n)=21*n^2 + 2, n >= 1.
%K A010011 nonn
%O A010011 0,2
%A A010011 njas
%I A044210
%S A044210 23,87,151,184,215,279,343,407,471,535,599,663,696,727,791,855,
%T A044210 919,983,1047,1111,1175,1208,1239,1303,1367,1431,1472,1559,1623,
%U A044210 1687,1720,1751,1815,1879,1943,2007,2071,2135,2199,2232,2263
%N A044210 String 2,7 occurs in the base 8 representation of n but not of n-1.
%K A044210 nonn,base
%O A044210 1,1
%A A044210 Clark Kimberling, ck6@cedar.evansville.edu
%I A044591
%S A044591 23,87,151,191,215,279,343,407,471,535,599,663,703,727,791,855,919,983,
%T A044591 1047,1111,1175,1215,1239,1303,1367,1431,1535,1559,1623,1687,1727,1751,
%U A044591 1815,1879,1943,2007,2071,2135,2199,2239,2263
%N A044591 String 2,7 occurs in the base 8 representation of n but not of n+1.
%K A044591 nonn,base
%O A044591 1,1
%A A044591 Clark Kimberling, ck6@cedar.evansville.edu
%I A050255
%S A050255 1,23,88,187,313,459,622,797,983,1179,1382,1592,1809,2031,2257,2489,
%T A050255 2724,2963,3205,3450,3698,3949,4203,4459,4717,4977,5239,5503,5768,6036,
%U A050255 6305,6575,6847,7121,7395,7671,7948,8227,8506,8787,9068,9351
%N A050255 Diaconis-Mosteller approximation to the Birthday problem function.
%D A050255 Diaconis, P. and Mosteller, F. "Methods of Studying Coincidences." J. Amer. Statist. Assoc. 84, 853-861, 1989.
%H A050255 E. W. Weisstein, Link to a section of The World of Mathematics.
%Y A050255 Cf. A014088, A050256.
%K A050255 nonn
%O A050255 1,2
%A A050255 Eric W. Weisstein (eric@weisstein.com)
%I A014088
%S A014088 1,23,88,187,313,460,623,798,985,1181,1385,1596,1813,2035,2263
%N A014088 Minimal number of people to give a 50% probability of having at least n coincident birthdays in one year.
%D A014088 P. Diaconis and F. Mosteller, Methods of studying coincidences, J. Amer. Statist. Assoc. 84 (1989) 853-861.
%H A014088 For more info
%H A014088 E. W. Weisstein, Link to a section of The World of Mathematics.
%K A014088 nonn
%O A014088 1,2
%A A014088 sfinch@mathsoft.com (Steven Finch)
%I A050529
%S A050529 23,89,353,1409,5767169,23068673,96757023244289,26596368031521841843535873,
%T A050529 467888254516290387262140085218681290753,1871553018065161549048560340874725163009,
%U A050529 9050275065266633231852330504065427777405047260984689248417349633
%N A050529 Primes of form 11*2^n+1.
%Y A050529 For more terms see A002261.
%K A050529 nonn
%O A050529 0,1
%A A050529 njas, Dec 29 1999
%I A042030
%S A042030 23,93,116,209,325,534,859,3970,183479,737886,921365,1659251,
%T A042030 2580616,4239867,6820483,31521799,1456823237,5858814747,
%U A042030 7315637984,13174452731,20490090715,33664543446,54154634161
%N A042030 Numerators of continued fraction convergents to sqrt(539).
%Y A042030 Cf. A042031.
%K A042030 nonn,cofr,easy
%O A042030 0,1
%A A042030 njas
%I A042032
%S A042032 23,93,395,488,5275,5763,28327,119071,5505593,22141443,
%T A042032 94071365,116212808,1256199445,1372412253,6745848457,28355806081,
%U A042032 1311112928183,5272807518813,22402343003435,27675150522248
%N A042032 Numerators of continued fraction convergents to sqrt(540).
%Y A042032 Cf. A042033.
%K A042032 nonn,cofr,easy
%O A042032 0,1
%A A042032 njas
%I A044274
%S A044274 23,104,185,207,266,347,428,509,590,671,752,833,914,936,995,1076,
%T A044274 1157,1238,1319,1400,1481,1562,1643,1665,1724,1805,1863,1967,
%U A044274 2048,2129,2210,2291,2372,2394,2453,2534,2615,2696,2777,2858
%N A044274 String 2,5 occurs in the base 9 representation of n but not of n-1.
%K A044274 nonn,base
%O A044274 1,1
%A A044274 Clark Kimberling, ck6@cedar.evansville.edu
%I A044655
%S A044655 23,104,185,215,266,347,428,509,590,671,752,833,914,944,995,1076,1157,
%T A044655 1238,1319,1400,1481,1562,1643,1673,1724,1805,1943,1967,2048,2129,2210,
%U A044655 2291,2372,2402,2453,2534,2615,2696,2777,2858
%N A044655 String 2,5 occurs in the base 9 representation of n but not of n+1.
%K A044655 nonn,base
%O A044655 1,1
%A A044655 Clark Kimberling, ck6@cedar.evansville.edu
%I A057877
%S A057877 23,113,1531,12239,111317,1111219,11119291,111111197,1111113173
%N A057877 First prime > 10^n which remains prime (no leading zeros allowed) after deleting all occurrences of its digits d.
%e A057877 1531 gives primes 53, 131 and 151 after dropping digits 1, 5 and 3.
%Y A057877 Cf. A057876-A057883, A051362, A034302-A034305.
%K A057877 nonn,base,more
%O A057877 1,1
%A A057877 Patrick De Geest (pdg@worldofnumbers.com), Oct 2000.
%I A042026
%S A042026 23,116,139,533,1205,1738,2943,4681,12305,16986,250109,
%T A042026 267095,784299,1051394,1835693,2887087,7609867,25716688,
%U A042026 33326555,192349463,8881401853,44599358728,53480760581
%N A042026 Numerators of continued fraction convergents to sqrt(537).
%Y A042026 Cf. A042027.
%K A042026 nonn,cofr,easy
%O A042026 0,1
%A A042026 njas
%I A042028
%S A042028 23,116,835,951,1786,13453,69051,3189799,16018046,115316121,
%T A042028 131334167,246650288,1857886183,9536081203,440517621521,
%U A042028 2212124188808,15925386943177,18137511131985,34062898075162
%N A042028 Numerators of continued fraction convergents to sqrt(538).
%Y A042028 Cf. A042029.
%K A042028 nonn,cofr,easy
%O A042028 0,1
%A A042028 njas
%I A044355
%S A044355 23,123,223,230,323,423,523,623,723,823,923,1023,1123,1223,1230,
%T A044355 1323,1423,1523,1623,1723,1823,1923,2023,2123,2223,2230,2300,
%U A044355 2423,2523,2623,2723,2823,2923,3023,3123,3223,3230,3323,3423
%N A044355 String 2,3 occurs in the base 10 representation of n but not of n-1.
%K A044355 nonn,base
%O A044355 1,1
%A A044355 Clark Kimberling, ck6@cedar.evansville.edu
%I A044736
%S A044736 23,123,223,239,323,423,523,623,723,823,923,1023,1123,1223,1239,1323,
%T A044736 1423,1523,1623,1723,1823,1923,2023,2123,2223,2239,2399,2423,2523,2623,
%U A044736 2723,2823,2923,3023,3123,3223,3239,3323,3423
%N A044736 String 2,3 occurs in the base 10 representation of n but not of n+1.
%K A044736 nonn,base
%O A044736 1,1
%A A044736 Clark Kimberling, ck6@cedar.evansville.edu
%I A033211
%S A033211 23,127,137,151,233,239,281,359,431,449,487,673,743,751,
%T A033211 911,953,967,977,1033,1087,1103,1129,1303,1409,1423,1439,
%U A033211 1481,1663,1759,1871,1873,2017,2039,2081,2129,2137,2207
%N A033211 Primes of form x^2+14*y^2.
%D A033211 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.
%K A033211 nonn
%O A033211 0,1
%A A033211 njas
%I A039613
%S A039613 23,133,155,179,191,203,215,227,239,251,263,275,276,278,279,280,281,
%T A039613 282,283,284,285,286,311,421,455,565,599,709,743,853,887,997,1031,1141,
%U A039613 1175,1285,1319,1429,1463,1573,1585,1596,1598,1599,1600,1601
%N A039613 Representation in base 12 has same nonzero number of 1's and 11's.
%K A039613 nonn,base,easy
%O A039613 0,1
%A A039613 Olivier Gerard (ogerard@ext.jussieu.fr)
%I A009482
%S A009482 1,1,23,135,4785,102353,4499623,167774711,9259741409,697193437985,
%T A009482 45391655836087,4778091975791591,356972130348630673,
%U A009482 60384518907092417009,5135539171238913723207
%V A009482 1,1,-23,-135,4785,102353,-4499623,-167774711,9259741409,697193437985,
%W A009482 -45391655836087,-4778091975791591,356972130348630673,
%X A009482 60384518907092417009,-5135539171238913723207
%N A009482 Expansion of sin(sin(x).cosh(x)).
%t A009482 Sin[ Sin[ x ]*Cosh[ x ] ] (* Odd Part *)
%K A009482 sign,done
%O A009482 0,3
%A A009482 R. H. Hardin (rhh@research.bell-labs.com)
%E A009482 Extended with signs 03/97 by Olivier Gerard.
%I A057883
%S A057883 23,137,6173,37019,5600239,476710937,8192454631
%N A057883 Smallest possible prime with at least n (from 2 to 10) distinct digits that remains prime (leading zeros not allowed) when all occurrences of its digits d are deleted.
%C A057883 Solutions for 8, 9 and 10 distinct digits not known yet.
%H A057883 C. Rivera, PrimePuzzle 110
%H A057883 C. K. Caldwell, Prime Curios! 5600239
%H A057883 C. K. Caldwell, Prime Curios! 476710937.
%e A057883 E.g. 5600239 is solution for at least 6 digits because 56239, 560039, 560029, 600239, 500239 and 560023 are all primes.
%Y A057883 Cf. A057876-A057882, A051362, A034302-A034305.
%K A057883 nonn,base,fini,nice,more
%O A057883 2,1
%A A057883 Patrick De Geest (pdg@worldofnumbers.com), Oct 2000.
%I A042024
%S A042024 23,139,162,301,764,4121,9006,13127,22133,145925,6734683,
%T A042024 40554023,47288706,87842729,222974164,1202713549,2628401262,
%U A042024 3831114811,6459516073,42588211249,1965517233527,11835691612411
%N A042024 Numerators of continued fraction convergents to sqrt(536).
%Y A042024 Cf. A042025.
%K A042024 nonn,cofr,easy
%O A042024 0,1
%A A042024 njas
%I A059915
%S A059915 23,139,211,422,461,761
%N A059915 A sequence f(n) of positive integers is called an F-sequence (in memory of Fibonacci) if it satisfies f(0)=0, f(1)=1, f(2)=2, and for all n > 2, either f(n) = f(n-1) + f(n-2) or f(n) = f(n-1) + f(n-3). A positive integer is called an F-number if it occurs in any F-sequence. Sequence gives numbers which are not F-numbers.
%C A059915 The sequence given above contains all non-F-numbers up to 5000000 (according to Klaus Nagel (nagel.klaus@t-online.de)).
%e A059915 22 IS an F-number because 0,1,2,2,3,5,7,10,15,22,... is an F-sequence. All Fibonacci-numbers are F-numbers.
%K A059915 hard,nonn
%O A059915 0,1
%A A059915 Christian Wieschebrink (wieschebrink@t-online.de), Feb 28 2001
%I A059701
%S A059701 23,139,239,241,409,431,577,647,761,787,1031,1051,1187,1319,1567,1831,
%T A059701 1879,2069,2087,2281,2347,2351,2521,3061,3167,3347,3433,4049,4231,4283,
%U A059701 4363,4447,4999,5437,5669,5689,5869,7499,7607,7681,7687,7841,7907,8039
%N A059701 Primes p such that p^8 reversed is also prime.
%t A059701 Select[ Range[ 11000 ], PrimeQ[ # ] && PrimeQ[ ToExpression[ StringReverse[ ToString[ #^8 ] ] ] ] & ]
%Y A059701 Cf. A059209.
%K A059701 base,nonn
%O A059701 1,1
%A A059701 Robert G. Wilson v (rgwv@kspaint.com), Feb 06 2001
%I A036494
%S A036494 0,1,23,140,512,1398,3175,6352,11585,19683,31623,48559,71832,102978,
%T A036494 143740,196070,262144,344366,445375,568056,715542,891224,1098758,
%U A036494 1342070,1625364,1953125,2330130,2761448,3252454,3808825,4436553
%N A036494 Nearest integer to n^(9/2).
%K A036494 nonn
%O A036494 0,3
%A A036494 njas
%I A055827
%S A055827 1,23,144,848,4880,27816,157920,895264,5074272,28772280,163262704,
%T A055827 927203184,5270629104,29988361032,170780080320,973422085184,
%U A055827 5552990609344,31702646247768,181128948471888,1035584204252560
%N A055827 T(2n+3,n), array T as in A055818.
%K A055827 nonn
%O A055827 0,2
%A A055827 Clark Kimberling, ck6@cedar.evansville.edu, May 28 2000
%I A037068
%S A037068 23,152,1107,8611,70478,1793210,5156463,45470645,2074530409,
%T A037068 11397691034,33578243459,639583252186
%N A037068 a(n)-th prime is the smallest prime containing exactly n 8's.
%H A037068 A. Booker, The Nth Prime Page
%Y A037068 Cf. A037069, A034388.
%K A037068 nonn,base,hard
%O A037068 1,1
%A A037068 Patrick De Geest (pdg@worldofnumbers.com), Jan 1999.
%I A042022
%S A042022 23,162,185,532,2313,2845,10848,13693,65620,144933,210553,
%T A042022 1618804,74675537,524347563,599023100,1722393763,7488598152,
%U A042022 9210991915,35121573897,44332565812,212451837145,469236240102
%N A042022 Numerators of continued fraction convergents to sqrt(535).
%Y A042022 Cf. A042023.
%K A042022 nonn,cofr,easy
%O A042022 0,1
%A A042022 njas
%I A062640
%S A062640 23,193,811
%N A062640 k^n - (k-1)^n is prime, where k is 74.
%C A062640 Terms > 1000 are often only strong pseudo-primes.
%Y A062640 Cf. A000043, A057468, A059801, A059802, A062572-A062666.
%K A062640 nonn,hard
%O A062640 0,1
%A A062640 Mike Oakes (Mikeoakes2@aol.com), May 18 2001, May 19 2001
%I A058193
%S A058193 23,199,523,1669,4297,9551,16141,28229,35617,43331,162143,31397,188029,
%T A058193 461717,404851,360653,1444309,2238823,492113,1895359,1671781,1357201,
%U A058193 3826019,11981443,13626257,17983717,39175217,37305713,52721113
%N A058193 Smallest prime p such that there is a gap of 6n between p and next prime.
%H A058193 T. R. Nicely, List of prime gaps
%F A058193 a(n)=A000230(3n)
%e A058193 d=72 appears after 31397, while smaller d=54,60,66 come later, following primes 35617,43331,162143 respectively
%Y A058193 A000230.
%K A058193 nonn
%O A058193 0,1
%A A058193 Labos E. (labos@ana1.sote.hu), Nov 28 2000
%I A065314
%S A065314 23,199,2297,30013,41,9699667,2819,53,21701,79,163,181,61,1619,14669,
%T A065314 307,103,306091,907,3217644767340672907899084554047,
%U A065314 267064515689275851355624017992701,23768741896345550770650537601358213
%N A065314 Least prime divisor of n-th primorial - (n+1)-st prime.
%F A065314 a(n)=A020639[A002110(n)-A000040(n+1)]
%e A065314 n=3, 3rd primorial=30,p(4)=7,difference=23, a(3)=23.
%Y A065314 Cf. A002110, A000040, A006530, A006530, A057713, A002584, A002585, A051342 A065314-A065317.
%K A065314 nonn
%O A065314 3,1
%A A065314 Labos E. (labos@ana1.sote.hu), Oct 29 2001
%I A065316
%S A065316 23,199,2297,30013,12451,9699667,79139,122069683,9241993,77184383,
%T A065316 211941187,72280449346243,73629553,142226610221,131076443530861861,
%U A065316 382046844818915214929,1348764323657,1822793973448088839487
%N A065316 Largest prime divisor of n-th primorial - (n+1)-st prime.
%F A065316 a(n)=A006530[A002110(n)-A000040(n+1)]
%e A065316 n=3, 3rd primorial=30, 4t prime=7. digfference=23, a(3)=23.
%Y A065316 Cf. A002110, A000040, A006530, A006530, A057713, A002584, A002585, A051342 A065314-A065317.
%K A065316 nonn
%O A065316 3,1
%A A065316 Labos E. (labos@ana1.sote.hu), Oct 29 2001
%I A060882
%S A060882 1,1,1,23,199,2297,30013,510491,9699667,223092841,6469693199,
%T A060882 200560490093,7420738134769,304250263527167,13082761331669983,
%U A060882 614889782588491357,32589158477190044671,1922760350154212639009
%V A060882 -1,-1,1,23,199,2297,30013,510491,9699667,223092841,6469693199,
%W A060882 200560490093,7420738134769,304250263527167,13082761331669983,
%X A060882 614889782588491357,32589158477190044671,1922760350154212639009
%N A060882 n-th primorial (A002110) - next prime.
%H A060882 Hisanori Mishima, Factorizations of many number sequences
%H A060882 Hisanori Mishima, Factorizations of many number sequences
%Y A060882 Cf. A002110, A060881.
%K A060882 sign,done
%O A060882 0,4
%A A060882 njas, May 05 2001
%I A022683
%S A022683 1,23,207,782,276,12259,29578,42711,229057,93863,828023,
%T A022683 2014110,4727719,15059963,22586736,58481962,8654877,246061935,
%U A022683 463250567,671892192,1993509889,2171787581,5545061605,20642183588
%V A022683 1,-23,207,-782,-276,12259,-29578,-42711,229057,93863,-828023,
%W A022683 -2014110,4727719,15059963,-22586736,-58481962,-8654877,246061935,
%X A022683 463250567,-671892192,-1993509889,-2171787581,5545061605,20642183588
%N A022683 Expansion of Product (1-m*q^m)^23; m=1..inf.
%K A022683 sign,done
%O A022683 0,2
%A A022683 njas
%I A042020
%S A042020 23,208,855,1063,1918,43259,45177,88436,398921,3678725,
%T A042020 169620271,1530261164,6290664927,7820926091,14111591018,
%U A042020 318275928487,332387519505,650663447992,2935041311473,27066035251249
%N A042020 Numerators of continued fraction convergents to sqrt(534).
%Y A042020 Cf. A042021.
%K A042020 nonn,cofr,easy
%O A042020 0,1
%A A042020 njas
%I A027481
%S A027481 23,215,1035,3535,9730,23058,48930,95370,173745,299585,493493,782145,
%T A027481 1199380
%N A027481 Third diagonal of A027477.
%F A027481 Numerators of sequence a[ n,n-2 ] in (a[ i,j ])^2 where a[ i,j ] = s(i,j)/i! if j<=i, 0 if j>i
%K A027481 nonn
%O A027481 3,1
%A A027481 Olivier Gerard (ogerard@ext.jussieu.fr)
(end)
Sloane's Database of Integer Sequences, Part 55
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/