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ODD MULTIPERFECT NUMBERS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 1 / August 2013
- Published online by Cambridge University Press:
- 06 November 2012, pp. 56-63
- Print publication:
- August 2013
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- Article
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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.
