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ODD MULTIPERFECT NUMBERS

Part of: Elementary number theory

Published online by Cambridge University Press:  06 November 2012

SHI-CHAO CHEN  and
HAO LUO
SHI-CHAO CHEN*
Affiliation:
Institute of Contemporary Mathematics,Department of Mathematics and Information Sciences, Henan University, Kaifeng 475001, PR China (email: schen@henu.edu.cn)
HAO LUO
Affiliation:
Institute of Contemporary Mathematics,Department of Mathematics and Information Sciences, Henan University, Kaifeng 475001, PR China (email: luohao200681@126.com)
*
✉For correspondence; e-mail: schen@henu.edu.cn
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Abstract

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A natural number $n$ is called multiperfect or $k$-perfect for integer $k\ge 2$ if $\sigma (n)=kn$, where $\sigma (n)$ is the sum of the positive divisors of $n$. In this paper, we establish a theorem on odd multiperfect numbers analogous to Euler’s theorem on odd perfect numbers. We describe the divisibility of the Euler part of odd multiperfect numbers and characterise the forms of odd perfect numbers $n=\pi ^\alpha M^2$ such that $\pi \equiv \alpha ~({\rm mod}~8)$, where $\pi ^\alpha $ is the Euler factor of $n$. We also present some examples to show the nonexistence of odd perfect numbers of certain forms.

Keywords

odd multiperfect numbersEuler partnonexistence

MSC classification

Secondary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 1 , August 2013 , pp. 56 - 63
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

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