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

JFM 58.1038.02
Full Text:

References:

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