Odd harmonic numbers exceed $10^{24}$
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- by Graeme L. Cohen and Ronald M. Sorli PDF
- Math. Comp. 79 (2010), 2451-2460 Request permission
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 $n>10^{24}$.References
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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
- Received by editor(s): May 26, 2009
- Received by editor(s) in revised form: August 6, 2009
- Published electronically: April 9, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 79 (2010), 2451-2460
- MSC (2010): Primary 11A25, 11Y70
- DOI: https://doi.org/10.1090/S0025-5718-10-02337-9
- MathSciNet review: 2684375