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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.01.005
Wu, Bing-Ling 1; Chen, Yong-Gao 1

1 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China
@article{CRMATH_2018__356_2_129_0,
author = {Wu, Bing-Ling and Chen, Yong-Gao},
title = {On the denominators of harmonic numbers},
journal = {Comptes Rendus. Math\'ematique},
pages = {129--132},
publisher = {Elsevier},
volume = {356},
number = {2},
year = {2018},
doi = {10.1016/j.crma.2018.01.005},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2018.01.005/}
}
TY  - JOUR
AU  - Wu, Bing-Ling
AU  - Chen, Yong-Gao
TI  - On the denominators of harmonic numbers
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 129
EP  - 132
VL  - 356
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.01.005/
DO  - 10.1016/j.crma.2018.01.005
LA  - en
ID  - CRMATH_2018__356_2_129_0
ER  - 
%0 Journal Article
%A Wu, Bing-Ling
%A Chen, Yong-Gao
%T On the denominators of harmonic numbers
%J Comptes Rendus. Mathématique
%D 2018
%P 129-132
%V 356
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.01.005/
%R 10.1016/j.crma.2018.01.005
%G en
%F CRMATH_2018__356_2_129_0
Wu, Bing-Ling; Chen, Yong-Gao. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.01.005/

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

[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

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.