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
.
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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.