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IMPROVED UPPER BOUNDS FOR ODD MULTIPERFECT NUMBERS

Part of: Sequences and sets Elementary number theory

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

Chen, F.-J. and Chen, Y.-G., ‘On odd perfect numbers’, Bull. Aust. Math. Soc. 86 (2012), 510–514.Google Scholar
Dickson, L. E., ‘Finiteness of odd perfect and primitive abundant numbers with $n$-distinct prime factors’, Amer. J. Math. 35 (1913), 413–422.Google Scholar
Dris, J. A. B. and Luca, F., A note on odd perfect numbers, Matimyas Mat., to appear, arXiv:1103.1437v3.Google Scholar
Goto, T. and Ohno, Y., ‘Odd perfect numbers have a prime factor exceeding $1{0}^{8} $’, Math. Comp. 77 (263) (2008), 1859–1868.Google Scholar
Heath-Brown, D. R., ‘Odd perfect numbers’, Math. Proc. Cambridge Philos. Soc. 115 (1994), 191–1946.Google Scholar
Luca, F. and Pomerance, C., ‘On the radical of a perfect number’, New York J. Math. 16 (2010), 23–30.Google Scholar
Nielsen, P. P., ‘An upper bound for odd perfect numbers’, Integers 3 (2003), A14; 9 pages.Google Scholar
Nielsen, P. P., ‘Odd perfect numbers have at least nine prime factors’, Math. Comp. 76 (260) (2007), 2109–2126.Google Scholar
Ochem, P. and Rao, M., ‘Odd perfect numbers are greater than $1{0}^{1500} $’, Math. Comp. 81 (279) (2012), 1869–1877.Google Scholar
Pomerance, C., ‘Multiply perfect numbers, Mersenne primes and effective computability’, Math. Ann. 266 (1997), 195–206.Google Scholar