Prove that there is an infinite number of values of $n$ for which the biggest prime divisor $p$ of $2^{n}-1$ satisfies $p < 2^{n/2021}-1$

Question

Prove that there is an infinte number of positive integers $n>1$ for which the biggest prime divisor of $2^n-1$ is smaller than $2^\frac{n}{2021}-1$ (which may or may not be integral).

I tried to prove that there is an infinite amount of values of $k,$ such that $n=2^k$ satisfies the statement of the problem.I think that this might work because $2^{2^k}-1=\prod_{i=1}^{k-1}(1+2^{2^i})$

Now I need to prove that the biggest prime divisor of each term of the product is smaller than $2^\frac{2^k}{2021}-1$.

This is were I got stuck.