Niven's constant is the limiting average of the largest exponent in the prime factorization of a positive integer. 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 0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, ... (OEIS A051904), while the first few terms of are 0, 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 0, 1/2, 2/3, 1, 1, 1, 1, 5/4, 4/3, 13/10, 14/11, 5/4, 16/13, ... (OEIS A380266 and A380267).

In addition, the scaled difference has the limit

(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 0, 1/2, 2/3, 1, 1, 1, 1, 5/4, 4/3, 13/10, 14/11, 4/3, 17/13, ... (OEIS A380264 and A380265).

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, counting the initial term at position 1, at which the partial quotients 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). The displayed positions are one greater than the zero-based positions in OEIS A033152 and OEIS A033154.