Niven's constant

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In number theory, Niven's constant 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 each natural number n > 1, then Niven's constant is given by

where ζ(k) is the value of the Riemann zeta function at the point k (Niven, 1969).

In the same paper Niven also proved that

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

[edit] See also

• Niven numbers

[edit] References

• Niven, Ivan M. (August 1969). "Averages of Exponents in Factoring Integers". Proceedings of the American Mathematical Society 22 (2): 356-360. Retrieved on 2007-03-08.