Affine-related quadratic polynomials, an index pre-sieve and finite truncations of Bateman–Horn local factors: an experimental study

We introduce an index pre-sieve for integer polynomials g(n), defined via congruence conditions modulo a finite set of primes P, with the goal of excluding a priori residue classes that necessarily produce composite values. We show that the natural density of the filtered subset of indices is described by a finite product of local factors, and that this product coincides with a finite truncation of the local factors appearing in the Bateman–Horn heuristic associated with g.

We then specialize the construction to an affine-related quadratic pair,

d(n) = n² + n + 5 and N(n) = 6n² + 6n + 31,

and present large-scale experimental results. We show that (i) an enrichment relative to the crude model 1 / log x is already present even in the absence of explicit filters, for local reasons intrinsic to the polynomials; and (ii) applying an index pre-sieve produces an additional enrichment consistent with the progressive approximation, via finite truncations, of the Bateman–Horn infinite product.