Can anyone point to any resource that comprehensively discusses a methodology that will "batch" factorise the composite values of this polynomial as it runs through all integer values of n?

I am not looking for conjecture on the feasibility of this notion; just fact as to whether or not anything has been published.

If the question is too specific, then please consider the wider query as to whether or not any known sieve methodology is available that will "batch" factorise polynomials of this form (i.e n^2 -n + (?)) where "?" can take on any constant integer value as n ranges through a natural succession of progressive integer values?

Please note : resulting factorisation need not be construed as being confined to an exhaustive list of prime factors!

Your help is appreciated and may well be significant.

Thank you