On an arithmetic function considered by Pillai
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 pn. On considère la suite finie décroissante définie récursivement par n 1 =n, n i+1 =n i -p(n i ) et dont le dernier terme, n r , 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 nx en lesquels R(n) prend une valeur donnée k.

For every positive integer n let p(n) be the largest prime number pn. Given a positive integer n=n 1 , we study the positive integer r=R(n) such that if we define recursively n i+1 =n i -p(n i ) for i1, then n r 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.

Reçu le : 2007-11-25
Publié le : 2010-03-22
DOI : https://doi.org/10.5802/jtnb.695
@article{JTNB_2009__21_3_695_0,
     author = {Florian Luca and Ravindranathan Thangadurai},
     title = {On an arithmetic function considered by Pillai},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {695--701},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     doi = {10.5802/jtnb.695},
     zbl = {1201.11092},
     mrnumber = {2605540},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2009__21_3_695_0/}
}
Florian Luca; Ravindranathan Thangadurai. 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://jtnb.centre-mersenne.org/item/JTNB_2009__21_3_695_0/

[1] R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes - II. Proc. London Math. Soc., (3) 83 (2001), 532–562. | MR 1851081 | Zbl 1016.11037

[2] H. Cramér, On the order of magnitude of the differences between consecutive prime numbers. Acta. Arith., 2 (1936), 396–403. | Zbl 0015.19702

[3] H. Halberstam and H. E. Rickert, Sieve methods. Academic Press, London, UK, 1974. | Zbl 0298.10026

[4] G.  Hoheisel, Primzahlprobleme in der Analysis.   Sitzunsberichte  der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 33 (1930), 3–11.

[5] T. R. Nicely, Some Results of Computational Research in Prime Numbers. http://www.trnicely.net/

[6] S.  S.  Pillai, An arithmetical function concerning primes. Annamalai University J. (1930), 159–167.

[7] R. Sitaramachandra Rao, On an error term of Landau - II in “Number theory (Winnipeg, Man., 1983)”, Rocky Mountain J. Math. 15 (1985), 579–588. | MR 823269 | Zbl 0584.10027