On the composition of a certain arithmetic function

Abstract

Let $S(n)$ be the function which associates for each positive integer $n$ the smallest positive integer $k$ such that $n\mid k!$. In this note, we look at various inequalities involving the composition of the function $S(n)$ with other standard arithmetic functions such as the Euler function and the sum of divisors function. We also look at the values of $S(F_n)$ and $S(L_n)$, where $F_n$ and $L_n$ are the $n$th Fibonacci and Lucas numbers, respectively.