  We consider a certain left action by the monoid $SL_2(\mathbf{N}_0)$ on the
set of divisor pairs $\mathcal{D}_f := \{ (m, n) \in \mathbf{N}_0 \times
\mathbf{N}_0 : m \lvert f(n) \}$ 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 $\hat{F}_f: SL_2(\mathbf{N}_0)
\rightarrow \mathcal{D}_f$. We call such polynomials $\textit{enumerable}$. 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
  $$ \begin{cases} \mathcal{S}(4k) = 2\mathcal{S}(2k) - \mathcal{S}(k) \\
\mathcal{S}(4k+1) = 2\mathcal{S}(2k) + \mathcal{S}(2k+1) \\ \mathcal{S}(4k+2) =
2\mathcal{S}(2k+1) + \mathcal{S}(2k) \\ \mathcal{S}(4k+3) = 2\mathcal{S}(2k+1)
- \mathcal{S}(k) \\ \end{cases} $$ 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 $|\mathcal{S}^{-1}(\{n\})| = \tau(n^2 + 1)$ 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.
