On Even Perfect Numbers II

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 $n\mathrm{\mid }{\sigma }_k(n)$ if and only if $n$ is an even perfect number $\mathrm{\ne }{2}^{k-1}(2^k-1)$. Furthermore, if $n=2^{\alpha -1}{p}^{\beta -1}$ for some $\beta >1$, then $n\ |
\sigma_5(n)$ifandonlyif$n$isanevenperfectnumber$\neq 496$.