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All numbers whose positive divisors have integral harmonic mean up to  |
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| Authors: | T Goto S Shibata |
| Institution: | Graduate School of Mathematics, Kyushu University 33, Fukuoka 812-8581, Japan ; Faculty of Mathematics, Kyushu University 33, Fukuoka 812-8581, Japan |
| 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  . |
| Keywords: | Harmonic number perfect number Ore's conjecture |
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