Polynomials whose divisors are enumerated by SL₂(N₀)

We consider a certain left action by the monoid SL₂(N₀) on the set of divisor pairs Df := \ (m, n) ∈ N₀ × N₀ : m f(n) \ where f ∈ Z[x] is a polynomial with integer coefficients. We classify all polynomials in Z[x] for which this action extends to an invertible map F̂f: SL₂(N₀) → Df. We call such polynomials enumerable. One of these polynomials happens to be f(n) = n² + 1. It is a well-known conjecture that there exist infinitely many primes of the form p = n² + 1. We construct a sequence S on the naturals defined by the recursions $$ {cases} S(4k) = 2S(2k) - S(k) \\ S(4k+1) = 2S(2k) + S(2k+1) \\ S(4k+2) = 2S(2k+1) + S(2k) \\ S(4k+3) = 2S(2k+1) - S(k) \\ {cases} $$ with initial conditions $S(1) = 0$, $S(2) = 1$, $S(3) = 1$. $$\{ S(k) \}k ∈ N = \{0,1,1,2,3,3,2,3,7,8,5,5,8,7,3, ⋯ \}$$ $S$ is shown to have the properties $1.$ For all $n ∈ N_0$, we have $S(2^n) = S(2ⁿ⁺¹ - 1) = n$. $2.$ For all $n ∈ N_0$, the size of the fiber of $n$ under $S$ satisfies $|S⁻¹(\{n\})| = τ(n^2 + 1)$ where $τ$ is the divisor counting function. $3.$ For all $n ∈ N_0$, the integer $n^2 + 1$ is prime if and only if $S⁻¹(\{n\}) = \{2^n, 2ⁿ⁺¹ - 1\}$. $4.$ $S(k)$ is a $2$-regular sequence.