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 Zx is a polynomial with integer coefficients. We classify all polynomials in Zx for which this action extends to an invertible map Ff: 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) \₊ ₍ = \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 N₀, we have S (2ⁿ) = S (2^n+1 - 1) = n. 2. For all n N₀, the size of the fiber of n under S satisfies |S^-1 (\n\) | = (n² + 1) where is the divisor counting function. 3. For all n N₀, the integer n² + 1 is prime if and only if S^-1 (\n\) = \2ⁿ, 2^{n+1 - 1\}. 4. S (k) is a 2-regular sequence.