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

Full-text PDF

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 .

**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*, 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 MR**0027292 (10:284a)****12.**C. Pomerance,*On a problem of Ore: harmonic numbers*, unpublished manuscript (1973); see Abstract *709-A5,*Notices Amer.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.

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.