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

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

Published electronically:
June 19, 2003

MathSciNet review:
2034133

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Abstract | References | Similar Articles | Additional Information

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

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