On even perfect numbers
Volume 154 / 2018
Colloquium Mathematicum 154 (2018), 131-136
MSC: Primary 11A25.
DOI: 10.4064/cm7374-11-2017
Published online: 6 August 2018
Abstract
Let $n=2^{\alpha -1}p^{\beta -1}$, where $\alpha , \beta \gt 1$ are two integers and $p$ is an odd prime. We prove that $n\mid\sigma _3(n)$ if and only if $n$ is an even perfect number $\not =28$, where $\sigma _3(n)=\sum _{d|n}d^3$. This extends one result of Cai, Chen and Zhang (2015).
