SIMON-MARAIS 2025 B1

A function $f: \{1, 2, \ldots, n\} \rightarrow \{m+1, m+2, \ldots, m+n\}$, where $m$ and $n$ are positive integers, is said to be beautiful if

• $a$ divides $f(a)$ for all $a \in \{1, 2, \ldots, n\}$, and
• $f(a) \neq f(b)$ for all distinct elements $a$ and $b$ of $\{1, 2, \ldots, n\}$ with $\gcd(a, b) \neq 1$.

Determine all positive integers $n$ such that, for every positive integer $m$, there exists a beautiful function $f: \{1, 2, \ldots, n\} \rightarrow \{m+1, m+2, \ldots, m+n\}$.