Question 5
For each natural number $n$, define $f(1)=1$ and, if $n>1$ has prime factorization $n=p_1^{a_1}\cdots p_k^{a_k}$ with distinct primes $p_i$ and positive integer exponents $a_i$, set $$f(n)=1+a_1p_1+\cdots+a_kp_k.$$ Let $f_s(n)$ denote the result of applying $f$ exactly $s$ times. Prove that for every given natural number $a$, there is a natural number $s_0$ such that $f_s(a)+f_{s-1}(a)$ is independent of the integer $s>s_0$. Here natural numbers in the domain are positive, as in the supplied definition.