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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

All numbers whose positive divisors have integral harmonic mean up to $\mathbf{300}$


Authors: T. Goto and S. Shibata
Journal: Math. Comp. 73 (2004), 475-491
MSC (2000): Primary 11A25, 11Y70
Published electronically: June 19, 2003
MathSciNet review: 2034133
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Abstract: A positive integer $n$ is said to be harmonic when the harmonic mean $H(n)$ of its positive divisors is an integer. Ore proved that every perfect number is harmonic. No nontrivial odd harmonic numbers are known. In this article, the list of all harmonic numbers $n$ with $H(n) \le 300$ is given. In particular, such harmonic numbers are all even except $1$.


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Additional Information

T. Goto
Affiliation: Graduate School of Mathematics, Kyushu University 33, Fukuoka 812-8581, Japan
Address at time of publication: Department of Mathematics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
Email: tgoto@math.kyushu-u.ac.jp, goto_takeshi@ma.noda.tus.ac.jp

S. Shibata
Affiliation: Faculty of Mathematics, Kyushu University 33, Fukuoka 812-8581, Japan
Email: ma200019@math.kyushu-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01554-0
PII: S 0025-5718(03)01554-0
Keywords: Harmonic number, perfect number, Ore's conjecture
Received by editor(s): December 10, 2001
Received by editor(s) in revised form: July 17, 2002
Published electronically: June 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society