Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. R. P. Brent and G. L. Cohen, ``A new lower bound for odd perfect numbers'', Math. Comp. 53 (1989), 431-437. MR 968150 (89m:11008)
  • 2. R. P. Brent, G. L. Cohen and H. J. J. te Riele, ``Improved techniques for lower bounds for odd perfect numbers'', Math. Comp. 57 (1991), 857-868. MR 1094940 (92c:11004)
  • 3. D. Callan, ``Solution to Problem 6616'', Amer. Math. Monthly 99 (1992), 783-789. MR 1542194
  • 4. G. L. Cohen, ``Numbers whose positive divisors have small harmonic mean'', Math. Comp. 68 (1997), 857-868. MR 1397443 (97f:11007)
  • 5. G. L. Cohen, Odd Harmonic Numbers Exceed $ 10^{18}$, internal report, University of Technology, Sydney (2008).
  • 6. G. L. Cohen and R. M. Sorli, ``Harmonic seeds'', Fibonacci Quart. 36 (1998), 386-390; errata, 39 (2001), 4. MR 1657575 (99j:11002)
  • 7. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fourth edition, Oxford (1962). MR 0067125 (16:673c)
  • 8. M. Garcia, ``On numbers with integral harmonic mean'', Amer. Math. Monthly 61 (1954), 89-96. MR 0059291 (15:506d)
  • 9. T. Goto and S. Shibata, ``All numbers whose positive divisors have integral harmonic mean up to 300'', Math. Comp. 73 (2004), 475-491. MR 2034133 (2004j:11005)
  • 10. W. H. Mills, ``On a conjecture of Ore'', Proceedings of the 1972 Number Theory Conference, University of Colorado, Boulder (1972), 142-146. MR 0389737 (52:10568)
  • 11. O. Ore, ``On the averages of the divisors of a number'', Amer. Math. Monthly 55 (1948), 615-619. MR 0027292 (10:284a)
  • 12. C. Pomerance, On a problem of Ore: harmonic numbers, unpublished manuscript (1973); see Abstract *709-A5, Notices Amer. Math. Soc. 20 (1973), A-648.
  • 13. W. Sierpiński, Elementary Theory of Numbers, Warsaw (1964). MR 930670 (89f:11003)
  • 14. R. M. Sorli, Algorithms in the Study of Multiperfect and Odd Perfect Numbers, Ph.D. thesis, University of Technology, Sydney (2003); available at http://hdl.handle.net/2100/275.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11A25, 11Y70

Retrieve articles in all journals with MSC (2010): 11A25, 11Y70


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.

American Mathematical Society