Odd harmonic numbers exceed

Authors:
Graeme L. Cohen and Ronald M. Sorli

Journal:
Math. Comp. **79** (2010), 2451-2460

MSC (2010):
Primary 11A25, 11Y70

DOI:
https://doi.org/10.1090/S0025-5718-10-02337-9

Published electronically:
April 9, 2010

MathSciNet review:
2684375

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Abstract | References | Similar Articles | Additional Information

Abstract: A number is harmonic if , where and are the number of positive divisors of and their sum, respectively. It is known that there are no odd harmonic numbers up to . We show here that, for any odd number , . It follows readily that if is odd and harmonic, then for any prime power divisor of , and we have used this in showing that . We subsequently showed that for any odd number , , from which it follows that if is odd and harmonic, then with as before, and we use this improved result in showing that .

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Additional Information

**Graeme L. Cohen**

Affiliation:
Department of Mathematical Sciences, University of Technology, Sydney, Broadway, New South Wales 2007, Australia

**Ronald M. Sorli**

Affiliation:
Department of Mathematical Sciences, University of Technology, Sydney, Broadway, New South Wales 2007, Australia

Email:
ron.sorli@uts.edu.au

DOI:
https://doi.org/10.1090/S0025-5718-10-02337-9

Received by editor(s):
May 26, 2009

Received by editor(s) in revised form:
August 6, 2009

Published electronically:
April 9, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.