## On an arithmetic function considered by Pillai.(English)Zbl 1201.11092

Given an integer $$n> 1$$, let $$p(n)$$ be the largest prime number, that is $$\leq n$$. S. Pillai [Journal Annamalai Univ. 1, 159–167 (1932; JFM 58.1038.02)] defined recursively an arithmetic function $$R(n): n_1= n$$, $$n_{k+1}= n_k- p(n_k)$$ if $$n_k> 1$$; put $$R(n)= k$$ if $$n_k$$ is prime or $$1$$. The authors generalize estimations of Pillai. They show $$R(n): O(\log\log n)$$ and $$\#\{n\leq x: R(n)= k\}\asymp{x\over\log_k x}$$ for every fixed integer $$k\geq 1(\log_k x$$ is the iterated logarithm).

### MSC:

 11N56 Rate of growth of arithmetic functions

### Keywords:

Pillai function; growth of arithmetic function

JFM 58.1038.02
Full Text:

### References:

  R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes - II. Proc. London Math. Soc., (3) 83 (2001), 532-562. · Zbl 1016.11037  H. Cramér, On the order of magnitude of the differences between consecutive prime numbers. Acta. Arith., 2 (1936), 396-403. · Zbl 0015.19702  H. Halberstam and H. E. Rickert, Sieve methods. Academic Press, London, UK, 1974. · Zbl 0298.10026  G. Hoheisel, Primzahlprobleme in der Analysis. Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 33 (1930), 3-11.  T. R. Nicely, Some Results of Computational Research in Prime Numbers. http://www.trnicely.net/ · Zbl 0923.11018  S. S. Pillai, An arithmetical function concerning primes. Annamalai University J. (1930), 159-167.  R. Sitaramachandra Rao, On an error term of Landau - II in “Number theory (Winnipeg, Man., 1983)”, Rocky Mountain J. Math. 15 (1985), 579-588. · Zbl 0584.10027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.