All numbers whose positive divisors have integral harmonic mean up to $\mathbf {300}$
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- by T. Goto and S. Shibata PDF
- Math. Comp. 73 (2004), 475-491 Request permission
Abstract:
A positive integer $n$ is said to be harmonic when the harmonic mean $H(n)$ 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 $n$ with $H(n) \le 300$ is given. In particular, such harmonic numbers are all even except $1$.References
- D. Callan, Solution to Problem 6616, Amer. Math. Monthly 99 (1992), 783–789.
- G. L. Cohen, Numbers whose positive divisors have small integral harmonic mean, Math. Comp. 66 (1997), no. 218, 883–891. MR 1397443, DOI 10.1090/S0025-5718-97-00819-3
- G. L. Cohen and R. M. Sorli, Harmonic seeds, Fibonacci Quart. 36 (1998), no. 5, 386–390. MR 1657575
- 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
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- 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, DOI 10.1007/978-1-4899-3585-4
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- Solution to Problem $E3445,$ Amer. Math. Monthly 99 (1992), 795.
- C. Pomerance, On a problem of Ore: Harmonic numbers (unpublished typescript); see Abstract 709-A5, Notices Amer. Math. Soc. 20 (1973) A-648.
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
- Received by editor(s): December 10, 2001
- Received by editor(s) in revised form: July 17, 2002
- Published electronically: June 19, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp. 73 (2004), 475-491
- MSC (2000): Primary 11A25, 11Y70
- DOI: https://doi.org/10.1090/S0025-5718-03-01554-0
- MathSciNet review: 2034133