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AN UPPER BOUND FOR THE NUMBER OF ODD MULTIPERFECT NUMBERS

Published online by Cambridge University Press:  28 January 2013

PINGZHI YUAN
PINGZHI YUAN*
Affiliation:
School of Mathematics, South China Normal University, Guangzhou 510631, PR China
*
[email protected]
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Abstract

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A natural number $n$ is called $k$-perfect if $\sigma (n)= kn$. In this paper, we show that for any integers $r\geq 2$ and $k\geq 2$, the number of odd $k$-perfect numbers $n$ with $\omega (n)\leq r$ is bounded by $\left({\lfloor {4}^{r} { \mathop{ \log } \nolimits }_{3} 2\rfloor + r\atop r} \right){ \mathop{ \sum } \nolimits }_{i= 1}^{r} \left({\lfloor kr/ 2\rfloor \atop i} \right)$, which is less than ${4}^{{r}^{2} } $ when $r$ is large enough.

Keywords

odd perfect numbersk-perfect numbers
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 1 , February 2014 , pp. 1 - 4
Copyright
©2013 Australian Mathematical Publishing Association Inc. 

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