On the equation $\varphi(2+\varphi(x))=2^{y-1}$, where $\varphi(n)$ denotes the Euler's totient function

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  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:

 Křížek, and Luca and Somer, 17 Lectures on Fermat Numbers, CMS Books in Mathematics Springer-Verlag (2001).

• With a computer, if you do a loop $x=1,2,3,4,5,\ldots$, you find many cases where $\varphi(2+\varphi(x))$ is a power of two, i.e. solutions to your equation. I agree it looks like that sequence of $x$ is not in OEIS. Terms below $100$ (ignoring the $y \ge 2$ condition) are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 27, 29, 30, 31, 38, 46, 47, 51, 54, 58, 59, 62, 64, 67, 68, 79, 80, 94, 96 – Jeppe Stig Nielsen Feb 4 '18 at 14:39
• The number of solutions (still with $y$ any integer) with $x<10^8$ is $371$. The last two of these are $98251370$ and $98252840$. – Jeppe Stig Nielsen Feb 4 '18 at 15:18
• Many thanks for this great help, since I've calculated the first few terms, it is a surprise to me that there are only 371 soutions when $x$ runs over the segment $[1,10^8]$. Many thanks for your attention one more time @JeppeStigNielsen – user243301 Feb 4 '18 at 17:43
• @user243301: there are just $27$ powers of $2$ in the interval $[1,10^8]$ and the function $\varphi(n)$ is, loosely speaking, weakly contractive, so it does not look so surprising to me. – Jack D'Aurizio Feb 4 '18 at 19:21
• For $x<10^9$ the count of solutions is $426$. – Jeppe Stig Nielsen Feb 5 '18 at 22:19