What are some basic properties of the quotient $s_n\over M_p$ where $M_p$ is a Mersenne prime?

Question

I've been exploring the Lucas-Lehmer test for a while now. I already know the square of one Mersenne $(2^n-1)^2$ is $(2^{n-1}-1)(2^{n+1}-1)+2^{n-1}$. Today I'm looking to use the properties of the other divisor of $s_{n-2}$, such that I can start the Lucas-Lehmer test from last Mersenne primes test. I know a bit about the other divisors already, like because the Lucas-Lehmer test is done with numbers that are 2 mod 4 and 3 mod 4 that the other divisor when it exists is 2 mod 4, and depending on n ( when it's prime) either 2 or 3 mod 7, and such (I haven't built the full list of residue classes for other known Mersenne primes yet). My attempt is for things like: $$s_{n-2}=a\cdot M_n$$ to be able to change to the next steps mod other Mersenne numbers. Any help ( even if I can't understand it quite yet), will be appreciated.