Let k > 2 be a prime such that 2^k – 1 is a Mersenne prime. Let n = 2^α–1p, where α > 1 and p < 3 · 2^α–1 – 1 is an odd prime. Define σ_k(n) to be the sum of the kth powers of the positive divisors of n. Continuing the work of Cai et al. and Jiang, we prove that n | σ_k(n) if and only if n is an even perfect number other than 2^k–1(2^k – 1). Furthermore, if n = 2^α–1p^β–1 for some β > 1, then n | σ₅(n) if and only if n is an even perfect number other than 496.