Divisibility properties of hyperharmonic numbers
type: Item
instance of: scholarly article
title: Divisibility properties of hyperharmonic numbers
zbMATH Open document ID: 1413.11052
published in: Acta Mathematica Hungarica
publication date: 2018-03-14T00:00:00Z
review text: The \(n\)th hyperharmonic number of order \(r\) is defined recursively by \(h_n^{(r)}=\sum_{k=1}^n h_k^{(r-1)}\), where \(h_n^{(1)}=\sum_{k=1}^n k^{-1}\) is the \(n\)th harmonic number. The authors prove a series of results for hyperharmonic numbers in this long paper. For instance, a generalization of Wolstenholme's theorem for hyperharmonic numbers is proved. Using combinatorial arguments they analyze hyperharmonic numbers which are not divisible by a prime. In particular, this yields that all hyperharmonic numbers in their reduced fractional form are odd. Given an odd prime \(p\), they further prove that card\(\{(n,r)\in {\mathbb N} : 1\leq n,r\leq x, h_n^{(r)}\equiv 0\pmod{p}\}\gg_p x^{\tau(p)}\) with explicitly given \(\tau(p)\). Contrary to an open problem whether there are only finitely many harmonic numbers \(h_n\) which are divisible by given prime \(p\), they show there are infinitely many hyperharmonic numbers \(h_n^{(r)}\) which are divisible by \(p\). More precisely, if \(J_p^{(r)}=\{n, \in {\mathbb N} : h_n^{(r)}\equiv 0\pmod{p}\}\) and \(p\geq r\geq2\), \(p\) odd prime, then for the counting function of \(J_p^{(r)}\) we have \(J_p^{(r)}(x)\gg_p \log x\). For the rest of results we refer the interested reader to the original paper which ends with two conjectures.
reviewed by: Štefan Porubský
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 6850208
zbMATH Keywords: hyperharmonic number
zbMATH Keywords: harmonic number
zbMATH Keywords: congruence identity
zbMATH Keywords: Wolstenholme's theorem
author: Haydar Göral
author: Doğa Can Sertbaş
describes a project that uses: SageMath
MaRDI profile type: Publication
full work available at URL: s10474-017-0766-7
OpenAlex ID: W2766838749
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DOI: 10.1007/S10474-017-0766-7
description: scientific article; zbMATH DE number 6850208