Powerfree sums of proper divisors

Abstract

Let $s(n):= \sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $k\ge 2$, the equivalence \[ \text{$n$ is $k$th powerfree} \Longleftrightarrow \text{$s(n)$ is $k$th powerfree} \] holds almost always (meaning, on a set of asymptotic density $1$). We prove this for $k\ge 4$.