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All numbers whose positive divisors have integral harmonic mean up to 
Author(s):
T.
Goto;
S.
Shibata.
Journal:
Math. Comp.
73
(2004),
475-491.
MSC (2000):
Primary 11A25, 11Y70
Posted:
June 19, 2003
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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  .
References:
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- [2]
- G. L. Cohen, Numbers whose positive divisors have small integral harmonic mean, Math. Comp. 66 (1997), 883-891. MR 97f:11007
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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:
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
Posted:
June 19, 2003
Copyright of article:
Copyright
2003,
American Mathematical Society
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