VN-TST 1991 Problem 5

For every natural number $n$ we define $f(n)$ by the following rule: $f(1) = 1$ and for $n>1$ then $f(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k$ , where $n = p_1^{a_1} \cdots p_k^{a_k}$ is the canonical prime factorisation of $n$ ( $p_1, \ldots, p_k$ are distinct primes and $a_1, \ldots, a_k$ are positive integers). For every positive integer $s$ , let $f_s(n) = f(f(\ldots f(n))\ldots)$ , where on the right hand side there are exactly $s$ symbols $f$ . Show that for every given natural number $a$ , there is a natural number $s_0$ such that for all $s > s_0$ , the sum $f_s(a) + f_{s-1}(a)$ does not depend on $s$ .