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All harmonic numbers less than \(10^{14}\). (English) Zbl 1154.11004

“A positive integer \(n\) is said to be harmonic if the harmonic mean of its positive divisors \[ H(n)=\frac{n\tau(n)}{\sigma(n)} \] is an integer, where \(\tau(n)\) denotes the number of the positive divisors of \(n\).”
Properties of harmonic numbers, including their relationship to perfect numbers, have been studied by Ø. Ore [Am. Math. Mon. 55, 615–619 (1948; Zbl 0031.10903)]. In this paper, the authors provide the list of all harmonic numbers less than \(10^{14}\). Answering a question of G. L. Cohen and R. M. Sorli [Fibonacci Q. 36, No. 5, 386–390 (1998; Zbl 0948.11004)], the authors show that every harmonic number does not have a unique harmonic seed.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11Y70 Values of arithmetic functions; tables
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[1] Brillhart, J.; Lehmer, D. H.; Selfridge, J. L.; Tuckerman, B.; Wagstaff, S. S., Factorizations ofb^n ± 1,b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers, Contemp. Math., 1-13 (2002), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0659.10001
[2] Cohen, G. L., Numbers whose positive divisors have small integral harmonic mean, Math. Comp., 66, 883-891 (1997) · Zbl 0882.11002 · doi:10.1090/S0025-5718-97-00819-3
[3] Cohen, G. L.; Sorli, R. M., Harmonic seeds, Fibonacci Quart., 36, 386-390 (1998) · Zbl 0948.11004
[4] Garcia, M., On numbers with integral harmonic mean, Amer. Math. Monthly, 61, 89-96 (1954) · Zbl 0058.27502 · doi:10.2307/2307792
[5] Goto, T.; Shibata, S., All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comp., 73, 475-491 (2004) · Zbl 1094.11005 · doi:10.1090/S0025-5718-03-01554-0
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[7] Ore, O., On the averages of the divisors of a number, Amer. Math. Monthly, 55, 615-619 (1948) · Zbl 0031.10903 · doi:10.2307/2305616
[8] Sorli, R. M., Algorithms in the Study of Multiperfect and Odd Perfect Numbers, Ph.D. thesis (2003), Sydney: University of Technology, Sydney
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.