Niven's constant

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

[math]\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 }[/math]

where ζ is the Riemann zeta function.

In the same paper Niven also proved that

[math]\displaystyle{ \sum_{j=1}^n h(j) = n + c\sqrt{n} + o (\sqrt{n}) }[/math]

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

[math]\displaystyle{ c = \frac{\zeta(\frac{3}{2})}{\zeta(3)}, }[/math]

and consequently that

[math]\displaystyle{ \lim_{n\to\infty} \frac{1}{n}\sum_{j=1}^n h(j) = 1. }[/math]

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". http://mathworld.wolfram.com/NivensConstant.html. 
• OEIS sequence A033150 (Niven's constant)