On an arithmetic function considered by Pillai
Luca, Florian1 ; Thangadurai, Ravindranathan2
1 Mathematical Institute UNAM, Ap. Postal 61-3 (Xangari), CP 58089 Morelia, Michoacán, Mexico
2 Harish-Chandra Research Institute Chhatnag Road, Jhunsi Allahabad 211 019, India
Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 695-701.

Soit n un nombre entier positif et p(n) le plus grand nombre premier p≤n. On considère la suite finie décroissante définie récursivement par n1=n, ni+1=ni-p(ni) et dont le dernier terme, nr, est soit premier soit égal à 1. On note R(n)=r la longueur de cette suite. Nous obtenons des majorations pour R(n) ainsi qu’une estimation du nombre d’éléments de l’ensemble des n≤x en lesquels R(n) prend une valeur donnée k.

For every positive integer n let p(n) be the largest prime number p≤n. Given a positive integer n=n1, we study the positive integer r=R(n) such that if we define recursively ni+1=ni-p(ni) for i≥1, then nr is a prime or 1. We obtain upper bounds for R(n) as well as an estimate for the set of n whose R(n) takes on a fixed value k.

DOI : 10.5802/jtnb.695
Luca, Florian 1 ; Thangadurai, Ravindranathan 2

1 Mathematical Institute UNAM, Ap. Postal 61-3 (Xangari), CP 58089 Morelia, Michoacán, Mexico
2 Harish-Chandra Research Institute Chhatnag Road, Jhunsi Allahabad 211 019, India 
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Luca, Florian; Thangadurai, Ravindranathan. On an arithmetic function considered by Pillai. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 695-701. doi : 10.5802/jtnb.695. https://www.numdam.org/articles/10.5802/jtnb.695/

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