Numbers whose positive divisors have small integral harmonic mean. (English) Zbl 0882.11002
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 studied by O. Ore [Am. Math. Mon. 55, 615-619 (1948; Zbl 0031.10903)]. M. Garcia [Am. Math. Mon. 61, 89-96 (1954; Zbl 0058.27502)] gives all harmonic numbers up to \(10^{7}\). In this paper, all harmonic numbers less than \(2\times 10^{9}\) are listed. R. K. Guy [Unsolved problems in number theory. 2nd ed. Unsolved Problems in Intuitive Mathematics. 1. New York, Springer (1994; Zbl 0805.11001)] wrote: “Which values does the harmonic mean take? Presumably not \(4, 12, 16, 18, 20, 22, \ldots\); does it take the value 23?” In this paper, this problem is solved for the first two values. In fact, all harmonic numbers \(n\) with \(H(n)\leq 13\) are determined.
Reviewer: P.Haukkanen (Tampere)
MSC:
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 11Y70 | Values of arithmetic functions; tables |
References:
| [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. · Zbl 0882.11002 |
| [3] | M. Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly 61 (1954), 89-96. · Zbl 0058.27502 |
| [4] | 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. · Zbl 0805.11001 |
| [5] | 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. · Zbl 0444.10004 |
| [6] | 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. · Zbl 0322.10003 |
| [7] | O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly 55 (1948), 615-619. · Zbl 0031.10903 |
| [8] | C. Pomerance, On a problem of Ore: harmonic numbers, unpublished manuscript, 1973. |
| [9] | R. Steuerwald, Ein Satz über natürliche Zahlen \(N\) mit \(\sigma (N)=3N\), Arch. Math. 5 (1954), 449-451. · Zbl 0056.03805 |
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