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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Superharmonic numbers


Author: Graeme L. Cohen
Journal: Math. Comp. 78 (2009), 421-429
MSC (2000): Primary 11A25, 11Y70
Published electronically: September 5, 2008
MathSciNet review: 2448714
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Abstract: Let $ \tau(n)$ denote the number of positive divisors of a natural number $ n>1$ and let $ \sigma(n)$ denote their sum. Then $ n$ is superharmonic if $ \sigma(n)\mid n^k\tau(n)$ for some positive integer $ k$. We deduce numerous properties of superharmonic numbers and show in particular that the set of all superharmonic numbers is the first nontrivial example that has been given of an infinite set that contains all perfect numbers but for which it is difficult to determine whether there is an odd member.


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Additional Information

Graeme L. Cohen
Affiliation: Department of Mathematical Sciences, University of Technology, Sydney, Broadway, NSW 2007, Australia
Address at time of publication: 1201/95 Brompton Road, Kensington, NSW 2033, Australia
Email: g.cohen@bigpond.net.au

DOI: http://dx.doi.org/10.1090/S0025-5718-08-02147-9
PII: S 0025-5718(08)02147-9
Received by editor(s): April 12, 2007
Received by editor(s) in revised form: January 22, 2008
Published electronically: September 5, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.