Problem

Source: 2008 Indonesia TST stage 2 test 4 p4

Tags: number theory, functional equation, functional, functional equation in N



Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies (i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$. (ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.