ODD MULTIPERFECT NUMBERS

• Published in: Bulletin of the Australian Mathematical Society Vol. 88; no. 1; pp. 56 - 63
• Main Authors: CHEN, SHI-CHAO, LUO, HAO
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.08.2013
• Subjects: Theorems; nonexistence; primary 11A05; Euler part; odd multiperfect numbers
• ISSN: 0004-9727; 1755-1633
• DOI: 10.1017/S0004972712000858

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.