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IMPROVED UPPER BOUNDS FOR ODD MULTIPERFECT NUMBERS

Part of: Elementary number theory Sequences and sets

Published online by Cambridge University Press:  12 June 2013

YONG-GAO CHEN  and
CUI-E TANG
YONG-GAO CHEN*
Affiliation:
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China
CUI-E TANG
Affiliation:
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China
*
✉[email protected]
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Abstract

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In this paper, we prove that, if $N$ is a positive odd number with $r$ distinct prime factors such that $N\mid \sigma (N)$, then $N\lt {2}^{{4}^{r} - {2}^{r} } $ and $N{\mathop{\prod }\nolimits}_{p\mid N} p\lt {2}^{{4}^{r} } $, where $\sigma (N)$ is the sum of all positive divisors of $N$. In particular, these bounds hold if $N$ is an odd perfect number.

Keywords

odd perfect numbersmultiperfect numbersupper bounds

MSC classification

Secondary: 11A25: Arithmetic functions; related numbers; inversion formulas 11B83: Special sequences and polynomials
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 3 , June 2014 , pp. 353 - 359
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

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