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Niven's Constant


Given a positive integer m>1, let its prime factorization be written

 m=p_1^(a_1)p_2^(a_2)p_3^(a_3)...p_k^(a_k).
(1)

Define the functions h(n) and H(n) by h(1)=1, H(1)=1, and

h(m)=min(a_1,a_2,...,a_k)
(2)
H(m)=max(a_1,a_2,...,a_k).
(3)

The first few terms of h(m) 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 H(m) are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, ... (OEIS A051903).

NivensConstantMin

Then the average value of h(m) tends to

 lim_(n->infty)1/nsum_(m=1)^nh(m)=1.
(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).

NivensConstantMinScaled

In addition, the ratio

 lim_(n->infty)(sum_(m=1)^(n)h(m)-n)/(sqrt(n))=(zeta(3/2))/(zeta(3)),
(5)

where zeta(z) is the Riemann zeta function (Niven 1969).

NivensConstantMax

Niven (1969) also proved that

 lim_(n->infty)1/nsum_(m=1)^nH(m)=C,
(6)

where Niven's constant C is given by

 C=1+{sum_(j=2)^infty[1-1/(zeta(j))]}=1.705211...
(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).


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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."

Referenced on Wolfram|Alpha

Niven's Constant

Cite this as:

Weisstein, Eric W. "Niven's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NivensConstant.html

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