Mathematics of Computation

Author(s): G. L. Cohen.
Journal: Math. Comp. 66 (1997), 883-891.
MSC (1991): Primary 11A25, 11Y70
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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.

1.: D. Callan, Solution to Problem 6616, Amer. Math. Monthly 99 (1992), 783-789.
2.: 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.
3.: M. Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly 61 (1954), 89-96. MR 15:506d
4.: R. K. Guy, Unsolved Problems in Number Theory, second edition, Springer-Verlag, New York, 1994. MR 96e:11002
5.: P. Hagis, Jr, Outline of a proof that every odd perfect number has at least eight prime factors, Math. Comp. 35 (1980), 1027-1032. MR 81k:10004
6.: W. H. Mills, On a conjecture of Ore, Proceedings of the 1972 Number Theory Conference, University of Colorado, Boulder, 1972, pp. 142-146. MR 52:10568
7.: O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly 55 (1948), 615-619. MR 10:284a
8.: C. Pomerance, On a problem of Ore: harmonic numbers, unpublished manuscript, 1973.
9.: R. Steuerwald, Ein Satz über natürliche Zahlen mit $\sigma (N)=3N$ , Arch. Math. 5 (1954), 449-451. MR 16:113h

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