Divisibility of Divisor Functions of Even Perfect Numbers
Hùng Việt Chu
Department of Mathematics
University of Illinois at Urbana-Champaign
Champaign, IL 61820
Let k > 2 be a prime such that 2k – 1
is a Mersenne
prime. Let n = 2α–1p, where α > 1 and
p < 3 · 2α–1 – 1 is an odd prime. Define
σk(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 | σk(n)
if and only if n is an even perfect number other than
2k–1(2k – 1). Furthermore, if n
= 2α–1pβ–1 for some β > 1,
then n | σ5(n) if and only if n
is an even perfect number other than 496.
Full version: pdf,
(Concerned with sequences
Received October 20 2020; revised version received February 13 2021.
Published in Journal of Integer Sequences,
February 13 2021.
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