GeorgeRP wrote: Does there exist a function $f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ such that \[ f(ab) = f(a)b + af(b) \]for all $a,b \in \mathbb{Z}_{\geq 0}$ and $f(p) > p^p$ for every prime number $p$? (Here, $\mathbb{Z}_{\geq 0}$ denotes the set of non-negative integers.) Yes, for example : $f(0)=f(1)=0$ and $f(\prod p_i^{n_i})=\prod p_i^{n_i}\sum n_ip_i^{p_i}$