Odd multiperfect numbers

Authors

  • S.-C. Chen
  • H. Luo

Keywords:

odd multiperfect numbers, Euler part, nonexistence

Abstract

A natural number $n$ is called {\it multiperfect} or {\it$k$-perfect} for integer $k\ge2$ if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the positive divisors of $n$. In this paper, we establish a structure theorem on odd multiperfect numbers analogous as Euler's theorem on odd perfect numbers. We prove the divisibility of the Euler part of odd multiperfect numbers and characterize the forms of odd perfect numbers $n=\pi^\alpha M^2$ such that $\pi\equiv\alpha 8$, where $\pi^\alpha$ is the Euler factor of $n$. We also present some examples to show the nonexistence of odd perfect numbers as applications. 10.1017/S0004972712000858

Published

2013-07-23

Issue

Vol. 88 No. 1 (2013)

Section

Articles
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