On the distribution of the number of prime factors
of the \(k\)-fold iterate of various arithmetic functions

Abstract. Given an arithmetic function \(f:{\mathbb N}\to {\mathbb N}\), let the \(k\)-fold iterate of \(f\) be defined by \(f_0(n)=n\) and \(f_{k}(n)=f(f_{k-1}(n))\) for each integer \(k\ge 1\). Let \(\omega(1)=0\) and, for each integer \(n\ge 2\), let \(\omega(n)\) stand for the number of distinct prime factors of \(n\). Here, we examine the distribution of the functions \(\omega(f_k(n))\) for various arithmetic functions \(f\).