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Divisibility of Divisor Functions of Even Perfect Numbers
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Hùng Việt Chu

Department of Mathematics

University of Illinois at Urbana-Champaign

Champaign, IL 61820

USA

**Abstract:**

Let *k* > 2 be a prime such that 2^{k} – 1
is a Mersenne
prime. Let *n* = 2^{α–1}*p*, where α > 1 and
*p* < 3 · 2^{α–1} – 1 is an odd prime. Define
σ_{k}(*n*) to be the sum of the *k*th 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–1}*p*^{β–1} for some β > 1,
then *n* | σ_{5}(*n*) if and only if *n*
is an even perfect number other than 496.

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(Concerned with sequences
A000396
A181595
A271816
A341475.)

Received October 20 2020; revised version received February 13 2021.
Published in *Journal of Integer Sequences*,
February 13 2021.

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