In number theory, Niven's constant, named after Ivan Niven, is the largest exponent appearing in the prime factorization of any natural number n "on average". More precisely, if we define H(1) = 1 and H(n) = the largest exponent appearing in the unique prime factorization of a natural number n > 1, then Niven's constant is given by
lim n → ∞ 1 n ∑ j = 1 n H ( j ) = 1 + ∑ k = 2 ∞ ( 1 − 1 ζ ( k ) ) = 1.705211 … {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}H(j)=1+\sum _{k=2}^{\infty }\left(1-{\frac {1}{\zeta (k)}}\right)=1.705211\dots }
where ζ is the Riemann zeta function. In the same paper Niven also proved that
∑ j = 1 n h ( j ) = n + c n + o ( n ) {\displaystyle \sum _{j=1}^{n}h(j)=n+c{\sqrt {n}}+o({\sqrt {n}})}
where h(1) = 1, h(n) = the smallest exponent appearing in the unique prime factorization of each natural number n > 1, o is little o notation, and the constant c is given by
c = ζ ( 3 2 ) ζ ( 3 ) , {\displaystyle c={\frac {\zeta ({\frac {3}{2}})}{\zeta (3)}},}
and consequently that
lim n → ∞ 1 n ∑ j = 1 n h ( j ) = 1. {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}h(j)=1.}
References
Further reading Steven R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 2003
External links Weisstein, Eric W. "Niven's Constant". MathWorld. OEIS sequence A033150 (Niven's constant)