On Even Perfect Numbers II

Abstract

Let $k>2$ be a prime such that $2^k-1$ is a Mersenne prime. Let $n=2^{\alpha -1}p$, where $\alpha >1$ and $p<3\cdot 2^{\alpha -1}-1$ is an odd prime. Continuing the work of Cai et al. and Jiang, we prove that $n|{\sigma }_k(n)$ if and only if $n$ is an even perfect number $\ne 2^{k-1}(2^k-1)$. Furthermore, if $n=2^{\alpha -1}{p}^{\beta -1}$ for some $\beta >1$, then $n|{\sigma }_5(n)$ if and only if $n$ is an even perfect number $\ne 496$.

12 pages