On an arithmetic function considered by Pillai
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, p. 695-701

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.

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.

@article{JTNB_2009__21_3_695_0,
     author = {Luca, Florian and Thangadurai, Ravindranathan},
     title = {On an arithmetic function considered by Pillai},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     pages = {695-701},
     doi = {10.5802/jtnb.695},
     mrnumber = {2605540},
     zbl = {1201.11092},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_3_695_0}
}
Luca, Florian; Thangadurai, Ravindranathan. On an arithmetic function considered by Pillai. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 695-701. doi : 10.5802/jtnb.695. http://www.numdam.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