All numbers whose positive divisors have integral harmonic mean up to

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 | References | Similar Articles | Additional Information

Abstract: A positive integer is said to be *harmonic* when the harmonic mean 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 with is given. In particular, such harmonic numbers are all even except .

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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:
https://doi.org/10.1090/S0025-5718-03-01554-0

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