Given a positive integer m > 1, let its prime factorization be written
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(1) |
Define the functions h(n) and H(n) by 
,
,
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The first few terms of h(m) are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, ... (Sloane's A052409), while the first few terms of H(m) are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, ... (Sloane's A051903).
Then the average value of h(m) tends to
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(4) |
Here, the running average values are given by 1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, 5/4, 16/13, ... (Sloane's A086195 and A086196).
In addition, the ratio
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(5) |
where 
is the Riemann zeta function (Niven 1969).
Niven (1969) also proved that
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(6) |
where Niven's constant C is given by
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(7) |
(Sloane's A033150). Here, the running average values are given by 1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, 5/4, 17/13, ... (Sloane's A086197 and A086198).
The continued fraction of Niven's constant is 1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 4, 4, 8, 4, 1, ... (Sloane's A033151). The positions at which the digits 1, 2, ... first occur in the continued fraction are 1, 3, 10, 7, 47, 41, 34, 13, 140, 252, 20, ... (Sloane's A033152). The sequence of largest terms in the continued fraction is 1, 2, 4, 8, 11, 14, 29, 372, 559, ... (Sloane's A033153), which occur at positions 1, 3, 7, 13, 20, 35, 51, 68, 96, ... (Sloane's A033154).
Finch, S. R. "Niven's Constant." §2.6 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 112-115, 2003.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 41, 1983.
Niven, I. "Averages of Exponents in Factoring Integers." Proc. Amer. Math. Soc. 22, 356-360, 1969.
Sloane, N. J. A. Sequences A033150, A033151, A033152, A033153, A033154, A051903, A052409, A086195, A086196, A086197, and A086198 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.
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