Niven's Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Given a Positive Integer $m>1$, let its Prime Factorization be written

$$m={p_1}^{a_1}{p_2}^{a_2}{p_3}^{a_3}\cdots {p_k}^{a_k}.$$ (1)

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

$$h(m) = \min(a_1, a_2, \ldots, a_k)$$ (2)

$$H(m) = \max(a_1, a_2, \ldots, a_k).$$ (3)

Then

$$\lim_{n\to\infty} {1\over n}\sum_{m=1}^n h(m)=1$$ (4)

$$\lim_{n\to\infty} {\sum_{m=1}^n h(m)-n\over\sqrt{n}} = {\zeta({\textstyle{3\over 2}})\over \zeta(3)},$$ (5)

where $\zeta(z)$ is the Riemann Zeta Function (Niven 1969). Niven (1969) also proved that

$$\lim_{n\to\infty} {1\over n}\sum_{m=1}^n H(m)=C,$$ (6)

where

$$C=1+\left\{{\sum_{j=2}^\infty \left[{1-{1\over\zeta(j)}}\right]}\right\}=1.705221\ldots$$ (7)

(Sloane's A033150).

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).

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/niven/niven.html

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.

Plouffe, S. ``The Niven Constant.'' http://www.lacim.uqam.ca/piDATA/niven.txt.

Sloane, N. J. A. Sequences A033150, A033151, A033152, A033153, and A033154 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.