Powerfree sums of proper divisors

TL;DR

This paper investigates the relationship between the powerfreeness of integers and their sum of proper divisors, proving the conjecture for all powers greater than or equal to four.

Contribution

It establishes that for all integers $k ge 4$, the property of being $k$th powerfree is asymptotically equivalent for numbers and their sum of proper divisors.

Findings

Proves the conjecture for $k ge 4$

Shows asymptotic density 1 for the equivalence

Advances understanding of divisor sum properties

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 $k\ge 4$.