Powerfree sums of proper divisors
Volume 168 / 2022
Colloquium Mathematicum 168 (2022), 287-295
MSC: Primary 11N37; Secondary 11A25, 11N64.
DOI: 10.4064/cm8616-10-2021
Published online: 3 January 2022
Abstract
Let $s(n):= \sum _{d\,|\, n,\,d \lt n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $k\ge 2$, the equivalence \[ \text {$n$ is $k$th powerfree} \iff \text {$s(n)$ is $k$th powerfree} \] holds almost always (meaning, on a set of asymptotic density $1$). We prove this for $k\ge 4$.
