Numbers whose positive divisors have small integral harmonic mean

Cohen, G.

doi:10.1090/S0025-5718-97-00819-3

Numbers whose positive divisors have small integral harmonic mean
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by G. L. Cohen PDF

Math. Comp. 66 (1997), 883-891 Request permission

Abstract:

A natural number $n$ is said to be harmonic when the harmonic mean $H(n)$ of its positive divisors is an integer. These were first introduced almost fifty years ago. In this paper, all harmonic numbers less than $2\times 10^{9}$ are listed, along with some other useful tables, and all harmonic numbers $n$ with $H(n)\le 13$ are determined.

References

D. Callan, Solution to Problem 6616, Amer. Math. Monthly 99 (1992), 783–789.
G. L. Cohen, Numbers whose positive divisors have small harmonic mean, Research Report R94-8 (June 1994), School of Mathematical Sciences, University of Technology, Sydney.
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
Peter Hagis Jr., Outline of a proof that every odd perfect number has at least eight prime factors, Math. Comp. 35 (1980), no. 151, 1027–1032. MR 572873, DOI 10.1090/S0025-5718-1980-0572873-9
W. H. Mills, On a conjecture of Ore, Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 142–146. MR 0389737
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
C. Pomerance, On a problem of Ore: harmonic numbers, unpublished manuscript, 1973.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946