I don't know if was in the literature a characterization of the solutions for pair of integers $(x,y)$ of the equation $$\varphi(2+\varphi(x))=2^{y-1},\tag{1}$$ where $\varphi(n)$ denotes the Euler's totient function and we consider that our integers satisfy $x\geq 1$ and $y\geq 2$.
Since I've created the equation with the purpose to get a relationship with numbers related with constructible polygons, and Mersenne primes I've the following basic facts of our equation $(1)$.
Claim 1. Each Mersenne prime $x=2^p-1$ satisfies $(1)$. That is, if we know a Mersenne prime $x=2^p-1$ then $$(x,y)=(2^p-1,p)$$ is a solution of $(1)$.$\square$
Because we know the relationship of Fermat primes and the power of two (and the Euler's totient function) and constructible polygons, see if you want this Wikipedia, or from [1] as Theorem 4.3 and Theorem 4.5, one deduce next.
Claim 2. If $(x,y)$ is a solution of $(1)$ then $$2+\varphi(x)=2^k\cdot(\text{ a product of distinct Fermat primes}),\tag{2}$$ for some positive integer $k\geq 0$.$\square$
Question. A) Do you know if the equation $(1)$ and how get families of solutions of $(1)$ were in the literature? Please, then answer this question as a reference request and I try to search and read those propositions concerning the solutions of the equation $(1)$. B) In other case I would like to know if is it feasible to get such characterization? Can you find different sequences/families of solutions $(x',y')$ of our equation $(1)$? Can you provide us a remarkable computational fact to know what about the different solutions $(x,y)$ of the equation $(1)$? Many thanks.
Thus I understand that neither Claim 1 nor Claim 2 provide us a full characterization of the solutions of $(1)$. As I've said in the first paragraph I would like to know if this equation was in the literature (I've search in OEIS and I believe that there is no an entry about the sequence of $x$ in our pairs $(x,y)$; and I believe that the chain phi(2+phi(x)) isn't showed from the browser of the OEIS), I think that is a nice equation than maybe was studied. I doubt that it is possible to find a full characterization of the solutions of $(1)$, because I think it is a very difficult problem.
References:
[1] Křížek, and Luca and Somer, 17 Lectures on Fermat Numbers, CMS Books in Mathematics Springer-Verlag (2001).