Are the hyperharmonics integral? A partial answer via the small intervals containing primes

Amrane, Rachid Aït; Belbachir, Hacène

doi:10.1016/j.crma.2010.12.015

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Hyperharmonic Numbers: Starting With H (0)1

On the Noninteger Property1

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2024

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Cited by 3 publications

(2 citation statements)

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“…the hyperharmonic numbers of order r are never integers except when n = 1 . This conjecture was justified for a class of pairs (n, r) by Ait-Amrane and Belbachir [1,2] and Cereceda [8]. Very recently Göral and Sertbaş [14] extended the current results for large orders; considering primes in short intervals, they proved that almost all hyperharmonic numbers are not integers.…”

Section: Hyperharmonic Numbers: Starting With H (0)mentioning

confidence: 81%

See 1 more Smart Citation

Evaluation of Euler-like sums via Hurwitz zeta values

Dil¹,

Mező²,

Cenkci³

2017

Turk J Math

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In this paper we collect two generalizations of harmonic numbers (namely generalized harmonic numbers and hyperharmonic numbers) under one roof. Recursion relations, closed-form evaluations, and generating functions of this unified extension are obtained. In light of this notion we evaluate some particular values of Euler sums in terms of odd zeta values. We also consider the noninteger property and some arithmetical aspects of this unified extension.

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Section: Hyperharmonic Numbers: Starting With H (0)mentioning

confidence: 81%

“…Later this result was improved by Amrane and Belbachir [1,2] (see also Cereceda [8]) to some general class of the parameter r . All these results were further sharpened by Göral and Sertbaş [14].…”

Section: On the Noninteger Propertymentioning

confidence: 97%