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Comptes Rendus Mathématique
Volume 339, n° 2
pages 87-90 (juillet 2004)
Doi : 10.1016/j.crma.2004.04.021
Received : 3 February 2004 ;  accepted : 30 April 2004
Le nombre des diviseurs d'un entier dans les progressions arithmétiques
The number of divisors of an integer in arithmetic progressions

Abdallah  Derbal a * ,  Abdelhakim  Smati b
aDépartement de mathématiques, École normale supérieure, BP 92, Vieux Kouba, Alger, Algérie 
bLaco, UMR-CNRS 6090, université de Limoges, 123, avenue Albert Thomas, 87060 Limoges cedex, France 

*Auteur correspondant.
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Soit   la fonction nombre des diviseurs de l'entier naturel  , dans les progressions arithmétiques  , avec   et   premiers entre-eux, et soit   définie par :   Dans cette Note, nous étudions et donnons la structure des nombres  -hautement composés supérieurs qui généralisent ceux définis par S. Ramanujan. Nous prouvons que le maximum absolu de   est atteint sur ces nombres et nous le donnons explicitement pour   ; généralisant ainsi l' étude faite par Nicolas et Robin pour  . Pour citer cet article : A. Derbal, A. Smati, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Abstract

Let   be the function number of divisors of the integer  , in arithmetic progressions  , with   and   coprime, and let   defined as follows:   In this Note, we study and give the structure of  -superior, highly composite numbers, which generalize those defined by S. Ramanujan. We prove that   reaches its maximum among these numbers. We give it explicitly for  . This generalizes the study of Nicolas and Robin, in which the case   is treated. To cite this article: A. Derbal, A. Smati, C. R. Acad. Sci. Paris, Ser. I 339 (2004).




© 2004  Académie des sciences@@#104156@@

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