CMO 2017 Problem 2

Let $f$ be a function from the set of positive integers to itself such that, for every $n$, the number of positive integer divisors of $n$ is equal to $f(f(n))$. For example, $f(f(6))=4$ and $f(f(25))=3$. Prove that if $p$ is prime then $f(p)$ is also prime.