Type: Article

On values of d(n!)/m!, phi(n!)/m! and sigma(n!)/m!

Daniel Baczkowski; Michael Filaseta; Florian Luca; Ognian Trifonov;

Abstract:

For f one of the classical arithmetic functions d, phi and sigma, we establish constraints on the quadruples (n, m, a, b) of integers satisfying f(n!)/m! = a/b. In particular, our results imply that as nm tends to infinity, the number of distinct prime divisors dividing the product of the numerator and denominator of the fraction f(n!)/m!, when reduced, tends to infinity.