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.


11A25 Arithmetic functions; related numbers; inversion formulas
11Y70 Values of arithmetic functions; tables
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