Pretty weak guarantee. Let $n=\textstyle \prod_{1\le i\le k}p_i^{e_i}$, where $p_1<\cdots<p_k$ are distinct primes and $e_i\ge 1$. Note that the given condition asserts \[ 2022\le \prod_{i\le k}\left(1+\frac{1}{p_i}+\cdots+\frac{1}{p_i^{e_i}}\right)<\prod_{i\le k}\left(1+\frac{1}{p_i-1}\right). \]Using the trivial bound $p_i\ge i$ for $i\ge 2$, we get \[ \prod_{i\le k}\left(1+\frac{1}{p_i-1}\right)\le 2\cdot \frac32\cdot \frac43\cdots \cdot \frac{k}{k-1} = k, \]we get directly that $k\ge 2022$.

Here is my solution: https://calimath.org/pdf/BalticWay2022-17.pdf And I uploaded the solution with motivation to: https://youtu.be/sio9uP04BK4