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 n2. 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 n2. 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.