On the periodicity of an arithmetical function

Hong, Shaofang; Yang, Yujuan

doi:10.1016/j.crma.2008.05.019

Number Theory

On the periodicity of an arithmetical function
[Sur la périodicité d'une fonction arithmétique]

Présenté par : Christophe Soulé

Hong, Shaofang ¹ ; Yang, Yujuan ¹

¹ Mathematical College, Sichuan University, Chengdu 610064, PR China

Comptes Rendus. Mathématique, Tome 346 (2008) no. 13-14, pp. 717-721.

Résumé
Abstract

Soit $k ⩾ 0$ un entier, en étudiant le plus petit commun multiple de $k + 1$ entiers consécutifs Farhi a introduit la fonction arithmétique définie par $g_{k} (n) : = \frac{n (n + 1) \dots (n + k)}{ppcm (n, n + 1, \dots, n + k)}$ pour n entier positif. Farhi a démontré que $g_{k}$ 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 $g_{k}$ . Dans cette Note, nous démontrons pour commencer $g_{k} (1) | g_{k} (n)$ pour tout entier positif n. Puis, utilisant ce résultat, nous montrons que $ppcm (1, 2, \dots, k)$ est une période de $g_{k}$ pour tout entier positif k, ce qui améliore le résultat de Farhi.

Let $k ⩾ 0$ be an integer. When studying the least common multiple of $k + 1$ consecutive integers, Farhi introduced the arithmetical function $g_{k}$ defined for any positive integer n by $g_{k} (n) : = \frac{n (n + 1) \dots (n + k)}{lcm (n, n + 1, \dots, n + k)}$ . Farhi proved that $g_{k}$ is periodic and k! is a period of $g_{k}$ . Meanwhile Farhi raised an open problem determining the smallest positive period of $g_{k}$ . In this Note, we first show that $g_{k} (1) | g_{k} (n)$ for all positive integers n. Consequently, using this result, we show that for all positive integers k, $lcm (1, 2, \dots, k)$ is a period of $g_{k}$ , thus improving Farhi's result.

Reçu le : 2007-10-31
Accepté le : 2008-05-26
Publié le : 2008-07-01

DOI : 10.1016/j.crma.2008.05.019

Affiliations des auteurs :

Hong, Shaofang ¹ ; Yang, Yujuan ¹

¹ Mathematical College, Sichuan University, Chengdu 610064, PR China

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Hong, Shaofang; Yang, Yujuan. On the periodicity of an arithmetical function. Comptes Rendus. Mathématique, Tome 346 (2008) no. 13-14, pp. 717-721. doi : 10.1016/j.crma.2008.05.019. http://www.numdam.org/articles/10.1016/j.crma.2008.05.019/

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Cité par Sources :

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