We say that a set $S$ of positive integers is special when there exists a function $f : \mathbb{N}\to \mathbb{N}$ satisfying that: $\bullet$ $f(k)\in S, \forall k\in\mathbb{N}$ $\bullet$ No integer $k$ with $2\le k \le 2021$ can be written as $\frac{af(a)}{bf(b)}$ with $a,b\in \mathbb{N}$ Find the smallest positive integer $n$ such that the set $S = \{ 1, 2021, 2021^2, \ldots , 2021^n \}$ is special or prove that no such integer exists. Note: $\mathbb{N}$ represents the set $\{ 1, 2, 3, \ldots \}$