Niven's Constant

Given a positive integer , let its prime factorization be written

(1)

Define the functions  and  by , , and

			(2)
			(3)

The first few terms of  are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, ... (OEIS A051904), while the first few terms of  are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, ... (OEIS A051903).

Then the average value of  tends to

(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, ... (OEIS A086195 and A086196).

In addition, the ratio

(5)

where  is the Riemann zeta function (Niven 1969).

Niven (1969) also proved that

(6)

where Niven's constant  is given by

(7)

(OEIS 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, ... (OEIS 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, ... (OEIS 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, ... (OEIS A033152). The sequence of largest terms in the continued fraction is 1, 2, 4, 8, 11, 14, 29, 372, 559, ... (OEIS A033153), which occur at positions 1, 3, 7, 13, 20, 35, 51, 68, 96, ... (OEIS A033154).

REFERENCES:

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, A051904, A086195, A086196, A086197, and A086198 in "The On-Line Encyclopedia of Integer Sequences."