Comptes Rendus
Number Theory
On the periodicity of an arithmetical function
Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 717-721.

Let k0 be an integer. When studying the least common multiple of k+1 consecutive integers, Farhi introduced the arithmetical function gk defined for any positive integer n by gk(n):=n(n+1)(n+k)lcm(n,n+1,,n+k). Farhi proved that gk is periodic and k! is a period of gk. Meanwhile Farhi raised an open problem determining the smallest positive period of gk. In this Note, we first show that gk(1)|gk(n) for all positive integers n. Consequently, using this result, we show that for all positive integers k, lcm(1,2,,k) is a period of gk, thus improving Farhi's result.

Soit k0 un entier, en étudiant le plus petit commun multiple de k+1 entiers consécutifs Farhi a introduit la fonction arithmétique définie par gk(n):=n(n+1)(n+k)ppcm(n,n+1,,n+k) pour n entier positif. Farhi a démontré que gk est périodique et que k! en est une période. Dans le même temps Farhi a posé la question de déterminer la plus petite période de gk. Dans cette Note, nous démontrons pour commencer gk(1)|gk(n) pour tout entier positif n. Puis, utilisant ce résultat, nous montrons que ppcm(1,2,,k) est une période de gk pour tout entier positif k, ce qui améliore le résultat de Farhi.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.05.019
Shaofang Hong 1; Yujuan Yang 1

1 Mathematical College, Sichuan University, Chengdu 610064, PR China
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Shaofang Hong; Yujuan Yang. On the periodicity of an arithmetical function. Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 717-721. doi : 10.1016/j.crma.2008.05.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.05.019/

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Cited by Sources:

This work was supported partially by Program for New Century Excellent Talents in University Grant # NCET-06-0785.

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