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Comptes Rendus Mathématique
Volume 348, n° 15-16
pages 835-838 (août 2010)
Doi : 10.1016/j.crma.2010.07.018
Received : 21 December 2009 ;  accepted : 19 July 2010
Sur un problème de S. Ramanujan, II
On problem by S. Ramanujan, II
 

Abdelhakim Smati
XLIM-UMR CNRS 6172, université de Limoges, 123, avenue Albert-Thomas, 87060 Limoges cedex, France 

Résumé

Dans une Note publiée aux C. R. Acad. Sci. Paris en 2005 et portant le même titre, nous avons étudié le problème posé par S. Ramanujan en 1915 qui consiste à déterminer l'ordre maximum de la fonction   où   est la fonction nombre des diviseurs de l'entier n et nous avons amélioré les résultats connus. Dans la présente note, on résout le problème, en déterminant aux constantes près, l'ordre maximum de  , ainsi que celui plus général, de la k -ième itérée de  ,  ,  . Cette généralisation a été formulée et étudiée par Erdős et Kátai en 1969.

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Abstract

In a Note published in C. R. Acad. Sci. Paris in 2005, and having the same title, we have studied problem set by S. Ramanujan in 1915, which consists to find the maximal order of the function  , where   is the function number of divisors of a integer n and we have improved the known results. In this note, we solve the problem by determining, to within constants, the maximal order of   as well as the more general one of the iterated k -fold of  . This generalization was formulated and studied by Erdős and Kátai in 1969.

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