IT-CESENATICO 2020 Problem 5

Le $S$ be the set of positive integers greater than or equal to $2$ . A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions:

1. $f$ is surjective
2. $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$ )
3. For every $n\in S$ the number $f(n)$ is the product of $f(p)$ , where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$ ). Determine the maximum and the minimum possible value of $f(2020)$ , when $f$ varies among all italian functions.