Download PDFOpen PDF in browserDivisibility of \Sigma_k(N) by Even Perfect NumbersEasyChair Preprint 94459 pages•Date: December 11, 2022AbstractLet n = 2(α−1)p(β−1) be a positive integer, where α, β > 1 and p is a prime satisfying p < 2(α−1− ln α / ln 2) − 1. Let k > 2 be a prime such that 2k −1 is a Mersenne prime and σk(n) = ∑d|n dk be the sum of the kth power of positive divisors of n. Continuing the work of Chu [4], we prove that n divides σk(n) if and only if and only n is an even perfect number ≠ to 2(k−1) (2k − 1) for all k ≦ 31. Keyphrases: congruence arithmetic, divisor function, perfect numbers
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