On the denominators of harmonic numbers

Wu, Bing-Ling; Chen, Yong-Gao

doi:10.1016/j.crma.2018.01.005

Number theory

On the denominators of harmonic numbers
[Sur les dénominateurs des nombres harmoniques]

Présenté par : the Editorial Board

Wu, Bing-Ling ¹ ; Chen, Yong-Gao ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China

Comptes Rendus. Mathématique, Tome 356 (2018) no. 2, pp. 129-132.

Résumé
Abstract

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

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

Reçu le : 2017-10-23
Accepté le : 2018-01-12
Publié le : 2018-02-01

DOI : 10.1016/j.crma.2018.01.005

Affiliations des auteurs :

Wu, Bing-Ling ¹ ; Chen, Yong-Gao ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China

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     title = {On the denominators of harmonic numbers},
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     pages = {129--132},
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Wu, Bing-Ling; Chen, Yong-Gao. On the denominators of harmonic numbers. Comptes Rendus. Mathématique, Tome 356 (2018) no. 2, pp. 129-132. doi : 10.1016/j.crma.2018.01.005. http://archive.numdam.org/articles/10.1016/j.crma.2018.01.005/

Bibliographie
Cité par

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[2] Eswarathasan, A.; Levine, E. p-integral harmonic sums, Discrete Math., Volume 91 (1991) no. 3, pp. 249-257

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

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

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

Cité par 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.