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 | References | Similar Articles | Additional Information

Abstract: Let denote the number of positive divisors of a natural number and let denote their sum. Then is *superharmonic* if for some positive integer . 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

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.