Niven's Constant
Given a positive integer 
, let its prime factorization be written
 |
(1)
|
Define the functions 
and 
by 
, 
, and
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
![C=1+<span style=]() {sum_(j=2)^infty[1-1/(zeta(j))]}=1.705211... " src="https://mathworld.wolfram.com/images/equations/NivensConstant/NumberedEquation5.gif" style="height:49px; width:233px" /> |
(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."
