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Odd harmonic numbers exceed $ 10^{24}$


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: A number $ n>1$ is harmonic if $ \sigma(n)\mid n\tau(n)$, where $ \tau(n)$ and $ \sigma(n)$ are the number of positive divisors of $ n$ and their sum, respectively. It is known that there are no odd harmonic numbers up to $ 10^{15}$. We show here that, for any odd number $ n>10^6$, $ \tau(n)\le n^{1/3}$. It follows readily that if $ n$ is odd and harmonic, then $ n>p^{3a/2}$ for any prime power divisor $ p^a$ of $ n$, and we have used this in showing that $ n>10^{18}$. We subsequently showed that for any odd number $ n>9\cdot 10^{17}$, $ \tau(n)\le n^{1/4}$, from which it follows that if $ n$ is odd and harmonic, then $ n>p^{8a/5}$ with $ p^a$ as before, and we use this improved result in showing that $ n>10^{24}$.


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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.