Number theory
On the denominators of harmonic numbers
Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 129-132.

Let $Hn$ be the n-th harmonic number and let $vn$ be its denominator. It is well known that $vn$ is even for every integer $n≥2$. In this paper, we study the properties of $vn$. One of our results is: the set of positive integers n such that $vn$ is divisible by the least common multiple of $1,2,⋯,⌊n1/4⌋$ has density one. In particular, for any positive integer m, the set of positive integers n such that $vn$ is divisible by m has density one.

Soit $Hn$ le n-ième nombre harmonique et notons $vn$ son dénominateur. Il est bien connu que $vn$ est pair pour tout entier $n≥2$. Dans ce texte, nous étudions les propriétés de $vn$. Un de nos résultats montre que l'ensemble des entiers positifs n tels que $vn$ soit divisible par le plus petit commun multiple de $1,2,…,[n1/4]$ est de densité 1. En particulier, pour tout entier positif m, l'ensemble des entiers positifs n tels que $vn$ soit divisible par m est de densité 1.

Accepted:
Published online:
DOI: 10.1016/j.crma.2018.01.005

Bing-Ling Wu 1; Yong-Gao Chen 1

1 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China
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Bing-Ling Wu; Yong-Gao Chen. On the denominators of harmonic numbers. Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 129-132. doi : 10.1016/j.crma.2018.01.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.01.005/

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[3] C. Sanna On the p-adic valuation of harmonic numbers, J. Number Theory, Volume 166 (2016), pp. 41-46

[4] P. Shiu The denominators of harmonic numbers | arXiv

[5] B.-L. Wu; Y.-G. Chen On certain properties of harmonic numbers, J. Number Theory, Volume 175 (2017), pp. 66-86

Cited by Sources:

This work was supported by the National Natural Science Foundation of China (No. 11771211) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.