Let $s(n):= sum_{dmid n,~d<n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $kge 2$, 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 $kge 4$.
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