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October 2007 All Harmonic Numbers Less than $10^{14}$
Takeshi Goto, Katsuyuki Okeya
Japan J. Indust. Appl. Math. 24(3): 275-288 (October 2007).


A positive integer $n$ is said to be \textit{harmonic} if the harmonic mean $H(n)$ of its positive divisors is an integer. Ore proved that every perfect number is harmonic and conjectured that there exist no odd harmonic numbers greater than $1$. In this article, we give the list of all harmonic numbers up to $10^{14}$. In particular, we find that there exist no nontrivial odd harmonic numbers less than $10^{14}$.


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Takeshi Goto. Katsuyuki Okeya. "All Harmonic Numbers Less than $10^{14}$." Japan J. Indust. Appl. Math. 24 (3) 275 - 288, October 2007.


Published: October 2007
First available in Project Euclid: 17 December 2007

zbMATH: 1154.11004
MathSciNet: MR2374991

Keywords: harmonic number , Ore's conjecture , perfect number

Rights: Copyright © 2007 The Japan Society for Industrial and Applied Mathematics

Vol.24 • No. 3 • October 2007
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