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

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

Published electronically:
June 19, 2003

MathSciNet review:
2034133

Full-text PDF Free Access

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 .

**[1]**D. Callan,*Solution to Problem*6616, Amer. Math. Monthly**99**(1992), 783-789.**[2]**G. L. Cohen,*Numbers whose positive divisors have small integral harmonic mean*, Math. Comp.**66**(1997), no. 218, 883–891. MR**1397443**, https://doi.org/10.1090/S0025-5718-97-00819-3**[3]**G. L. Cohen and R. M. Sorli,*Harmonic seeds*, Fibonacci Quart.**36**(1998), no. 5, 386–390. MR**1657575****[4]**G. L. Cohen and Deng Moujie,*On a generalisation of Ore’s harmonic numbers*, Nieuw Arch. Wisk. (4)**16**(1998), no. 3, 161–172. MR**1680101****[5]**M. Garcia,*On numbers with integral harmonic mean,*Amer. Math. Monthly**61**(1954), 89-96. MR**15:506d****[6]**Richard K. Guy,*Unsolved problems in number theory*, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. MR**1299330****[7]**H. J. Kanold,*Über das harmonische Mittel der Teiler einer natürlichen Zahl,*Math. Ann.**133**(1957), 371-374. MR**19:635f****[8]**O. Ore,*On the averages of the divisors of a number,*Amer. Math. Monthly**55**(1948), 615-619. MR**10:284a****[9]***Solution to Problem*Amer. Math. Monthly**99**(1992), 795.**[10]**C. Pomerance,*On a problem of Ore: Harmonic numbers*(unpublished typescript); see Abstract 709-A5, Notices Amer. Math. Soc.**20**(1973) A-648.

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
11A25,
11Y70

Retrieve articles in all journals with MSC (2000): 11A25, 11Y70

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