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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

Citations:

JFM 58.1038.02
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References:

[1] 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
[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/ · Zbl 0923.11018
[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. · Zbl 0584.10027
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