All numbers whose positive divisors have integral harmonic mean up to
Authors:
T. Goto and S. Shibata
Journal:
Math. Comp. 73 (2004), 475491
MSC (2000):
Primary 11A25, 11Y70
Published electronically:
June 19, 2003
MathSciNet review:
2034133
Fulltext PDF Free Access
Abstract 
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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), 783789.
 [2]
G.
L. Cohen, Numbers whose positive divisors have
small integral harmonic mean, Math. Comp.
66 (1997), no. 218, 883–891. MR 1397443
(97f:11007), http://dx.doi.org/10.1090/S0025571897008193
 [3]
G.
L. Cohen and R.
M. Sorli, Harmonic seeds, Fibonacci Quart. 36
(1998), no. 5, 386–390. MR 1657575
(99j:11002)
 [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
(2000k:11008)
 [5]
Mariano
Garcia, On numbers with integral harmonic mean, Amer. Math.
Monthly 61 (1954), 89–96. MR 0059291
(15,506d)
 [6]
Richard
K. Guy, Unsolved problems in number theory, 2nd ed., Problem
Books in Mathematics, SpringerVerlag, New York, 1994. Unsolved Problems in
Intuitive Mathematics, I. MR 1299330
(96e:11002)
 [7]
HansJoachim
Kanold, Über das harmonische Mittel der Teiler einer
natürlichen Zahl, Math. Ann. 133 (1957),
371–374 (German). MR 0089219
(19,635f)
 [8]
Oystein
Ore, On the averages of the divisors of a number, Amer. Math.
Monthly 55 (1948), 615–619. MR 0027292
(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C. Pomerance, On a problem of Ore: Harmonic numbers (unpublished typescript); see Abstract 709A5, Notices Amer. Math. Soc. 20 (1973) A648.
 [1]
 D. Callan, Solution to Problem 6616, Amer. Math. Monthly 99 (1992), 783789.
 [2]
 G. L. Cohen, Numbers whose positive divisors have small integral harmonic mean, Math. Comp. 66 (1997), 883891. MR 97f:11007
 [3]
 G. L. Cohen and R. M. Sorli, Harmonic seeds, Fibonacci Quart. 36 (1998), 386390; Errata, Fibonacci Quart. 39 (2001), 4. MR 99j:11002
 [4]
 G. L. Cohen and Deng Moujie, On a generalisation of Ore's harmonic numbers, Nieuw. Arch. Wisk. (4) 16 (1998), 161172. MR 2000k:11008
 [5]
 M. Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly 61 (1954), 8996. MR 15:506d
 [6]
 R. K. Guy, Unsolved Problems in Number Theory, second edition, SpringerVerlag, New York, 1994. MR 96e:11002
 [7]
 H. J. Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann. 133 (1957), 371374. MR 19:635f
 [8]
 O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly 55 (1948), 615619. 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 709A5, Notices Amer. Math. Soc. 20 (1973) A648.
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Additional Information
T. Goto
Affiliation:
Graduate School of Mathematics, Kyushu University 33, Fukuoka 8128581, Japan
Address at time of publication:
Department of Mathematics, Tokyo University of Science, Noda, Chiba 2788510, Japan
Email:
tgoto@math.kyushuu.ac.jp, goto_takeshi@ma.noda.tus.ac.jp
S. Shibata
Affiliation:
Faculty of Mathematics, Kyushu University 33, Fukuoka 8128581, Japan
Email:
ma200019@math.kyushuu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0025571803015540
PII:
S 00255718(03)015540
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
