Polynomials whose divisors are enumerated by $S{L}_2(N_0)$

We consider a certain left action by the monoid $S{L}_2({\mathbf{N}}_0)$ on the set of divisor pairs ${\mathcal{D}}_f:=\{(m,n)\in {\mathbf{N}}_0\times {\mathbf{N}}_0:m\mid f(n)\}$ where $f\in \mathbf{Z}[x]$ is a polynomial with integer coefficients. We classify all polynomials in $\mathbf{Z}[x]$ for which this action extends to an invertible map ${\widehat{F}}_f:S{L}_2({\mathbf{N}}_0)\to {\mathcal{D}}_f$. We call such polynomials $\mathit{\text{enumerable}}$. One of these polynomials happens to be $f(n)=n^2+1$. It is a well-known conjecture that there exist infinitely many primes of the form $p=n^2+1$. We construct a sequence $\mathcal{S}$ on the naturals defined by the recursions $$\{\begin{array}{l}\mathcal{S}(4k)=2\mathcal{S}(2k)-\mathcal{S}(k)\\ \mathcal{S}(4k+1)=2\mathcal{S}(2k)+\mathcal{S}(2k+1)\\ \mathcal{S}(4k+2)=2\mathcal{S}(2k+1)+\mathcal{S}(2k)\\ \mathcal{S}(4k+3)=2\mathcal{S}(2k+1)-\mathcal{S}(k)\end{array}$$ with initial conditions $\mathcal{S}(1)=0$, $\mathcal{S}(2)=1$, $\mathcal{S}(3)=1$. $$\{\mathcal{S}(k){\}}_{k\in \mathbf{N}}=\{0,1,1,2,3,3,2,3,7,8,5,5,8,7,3,\cdots \}$$ $\mathcal{S}$ is shown to have the properties $1.$ For all $n\in {\mathbf{N}}_0$, we have $\mathcal{S}(2^n)=\mathcal{S}(2^{n+1}-1)=n$. $2.$ For all $n\in {\mathbf{N}}_0$, the size of the fiber of $n$ under $\mathcal{S}$ satisfies $\mathrm{\mid }{\mathcal{S}}^{-1}(\{n\})\mathrm{\mid }=\tau (n^2+1)$ where $\tau$ is the divisor counting function. $3.$ For all $n\in {\mathbf{N}}_0$, the integer $n^2+1$ is prime if and only if ${\mathcal{S}}^{-1}(\{n\})=\{2^n,2^{n+1}-1\}$. $4.$ $\mathcal{S}(k)$ is a $2$-regular sequence.