On Quotients of Values of Euler's Function on Factorials

Abstract

Recently, there has been some interest in values of arithmetical functions on members of special sequences, such as Euler's totient function ϕ on factorials, linear recurrences, etc. In this article, we investigate, for given positive integers a and b, the least positive integer c = c(a, b) such that the quotient ϕ(c!)/ϕ(a!)ϕ(b!) is an integer. We derive results on the limit of the ratio c(a, b)/(a + b) as a and b tend to infinity. Furthermore, we show that c(a, b) > a + b for all pairs of positive integers (a, b) with an exception of a set of density zero.