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 $n\ |\ \sigma_k(n)$ if and only if $n$ is an even perfect number $\neq 2^{k-1}(2^k-1)$. 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 $\neq 496$.