On the denominators of harmonic numbers

Bing-Ling Wu; Yong-Gao Chen

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

Bing-Ling Wu ¹ ; Yong-Gao Chen ¹

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

Comptes Rendus. Mathématique, Volume 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 :

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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     title = {On the denominators of harmonic numbers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {129--132},
     publisher = {Elsevier},
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     number = {2},
     year = {2018},
     doi = {10.1016/j.crma.2018.01.005},
     language = {en},
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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/

Bibliographie
Cité par

[1] D.W. Boyd A p-adic study of the partial sums of the harmonic series, Exp. Math., Volume 3 (1994) no. 4, pp. 287-302

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

[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

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.

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