When I was exploring congruences$\dagger$ involving the terms $S_k$ defined in Lucas-Lehmer test (this reference is the Wikipedia, but the main reference is Crandall and Pomerance, Prime Numbers: A Computational Perspective Springer(2001)) I've said to me that perhaps could be an interesting idea to combine relations that satisfies the sum of divisors function $\sigma(n)=\sum_{d\mid n}d$ and some of the terms of the sequence $S_k$ defined in the Lucas-Lehmer test.
I believe that is right and that can deduce easily (if my computations were rigths, from a easy manipulation of the identities $S_{p-1}=S^2_{p-2}-2$ and $S_{p}=S^2_{p-1}-2$) that on assumption that $M_p=2^p-1$ is a Mersenne prime (a fixed Mersenne prime) with $p>2$ then $$S_{p}\equiv 2\text{mod}M_p,$$ thus since $M_p$ is prime one has that $$\text{gcd}(S_{p},M_p)=1,$$ and from this, since the sum of divisors function is multiplicative then (here was fixed a mistake, thanks to the user in comments)$$\sigma(S_pM_p)=\sigma(S_p)2^p.$$
Question 1. Were rights my computations?
On assumption that $\text{gcd}(\sigma(S_{p}),M_p)=1$, also one could do a nested calculation $$\sigma(\sigma(S_pM_p)-\sigma(S_p))=\sigma(\sigma(S_p))2^p.$$
Question 2. What's about this condition $\text{gcd}(\sigma(S_{p}),M_p)=1$, when $M_p$ is a Mersenne prime with $p>2$? Can you find a counterexample for this condition?
Feel free if there are mistakes to do the corrections. Many thanks.
$\dagger$ Appendix: It is possible to get a lot of such conguences by reproducing the shape of a Mersenne prime, using the identity (identities) involving Lucas-Lehmer test for a fixed Mersenne prime $M_p$ with $p>2$, Euler-Fermat theorem, and the Newton binomial theorems for specials sums or products, and after dividing by $M_p=2^p-1$. You are welcome if you want to explore and ask about these simple and complicated identities.