Question 117
Let $S$ be the set of integers at least 2. A function $f:S\to S$ is called primordial if it has these properties:
- It is surjective: for every $s\in S$ there is at least one $n\in S$ such that $f(n)=s$.
- It is increasing on the primes: if primes $p_1\lt p_2$, then $f(p_1)\lt f(p_2)$.
- For every $n\in S$, $f(n)$ is the product of $f(p)$ over all distinct primes dividing $n$; for example, $f(360)=f(2^3\cdot3^2\cdot5)=f(2)f(3)f(5)$.
Determine the largest and smallest possible values of $f(2020)$ among all primordial functions.
- It is surjective: for every $s\in S$ there is at least one $n\in S$ such that $f(n)=s$.
- It is increasing on the primes: if primes $p_1\lt p_2$, then $f(p_1)\lt f(p_2)$.
- For every $n\in S$, $f(n)$ is the product of $f(p)$ over all distinct primes dividing $n$; for example, $f(360)=f(2^3\cdot3^2\cdot5)=f(2)f(3)f(5)$.
Determine the largest and smallest possible values of $f(2020)$ among all primordial functions.