A proof of Sondow's conjecture on the Smarandache function

Abstract

The Smarandache function of a positive integer $n$, denoted by $S(n)$, is defined to be the smallest positive integer $j$ such that $n$ divides the factorial $j!$. In this note, we prove that for any fixed number $k>1$, the inequality $n^k<S(n)!$ holds for almost all positive integers $n$. This confirms Sondow's conjecture which asserts that the inequality $n^2<S(n)!$ holds for almost all positive integers $n$.

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