Comment by Gro-Tsen on Can exist a positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ are not prime for all $n \ge 1$?

I understand the proof, but I'm confused as to what “makes it work”. How difficult is it to find $n$ such that we can cover the residue classes mod $n$ by arithmetic sequences with difference $r_i$ where $r_i$ is the multiplicative order of $2$ mod $p_i$ where $p_i$ are the prime factors of $2^n-1$ (if I correctly summarized what property of $n=64$ is being used here)? Do you have additional insight to share? (Maybe this should be a separate question.)