Rational Values Involving Arithmetic Functions Evaluated at Factorials
by
Dan Baczkowski
University of South Carolina
Coauthors: Michael Filaseta, Florian Luca, Ognian Trifonov

Florian Luca established that for a fixed rational number r, there are a finite number of positive integers n and m for which f(n!) = r ·m! where f is one of the arithmetic functions t (the number of divisors function), f (Euler's f-function), or s (the sum of the divisors function). In this joint work, we establish a generalization of these results, in particular a consequence of our work is the following: Let k be a fixed positive integer. Then there are finitely many positive integers n, m, a and b such that

b ·f(n!) = a ·m!,     gcd (a, b) = 1     and     w(a b) ≤ k

where w(·) denotes the number of distinct prime divisors function.

Date received: April 28, 2008