Divisibility of \Sigma_k(N) by Even Perfect Numbers

EasyChair Preprint 9445

Abstract

Let 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 2^k −1 is a Mersenne prime and σ_k(n) = ∑_d|n d^k be the sum of the k^th power of positive divisors of n. Continuing the work of Chu [4], we prove that n divides σ_k(n) if and only if and only n is an even perfect number ≠ to 2^(k−1) (2^k − 1) for all k ≦ 31.

Keyphrases: congruence arithmetic, divisor function, perfect numbers

@booklet{EasyChair:9445,
  author    = {Emmanuel Elima},
  title     = {Divisibility of \Sigma_k(N) by Even Perfect Numbers},
  howpublished = {EasyChair Preprint 9445},
  year      = {EasyChair, 2022}}