On an arithmetic function considered by Pillai (Q988071)

Given an integer \(n> 1\), let \(p(n)\) be the largest prime number, that is \(\leq n\). \textit{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).