For a positive integer $n$ let $\sigma(n)$ be the sum of divisors function of $n$. In this note, we fix a positive integer $a$ and we investigate the positive integers $n$ such that $n\mid a^{\sigma(n)}-1$. We also show that under a plausible hypothesis related to the distribution of prime numbers there exist infinitely many positive integers $n$ such that $n\mid a^{\sigma(n)}-1$ holds for all integers a coprime to $n$.