[math-fun] Polynomial primogeniture

Just for fun, I decided to look at the factorization of Q(N) = N^2+N+1 as N gets large. So far I've just looked at 1 <= N <= 1000, but there appears to be an unusually high frequency of Q(N)'s being prime. And also when Q(N) is composite, it appears as though there tend to be unusually few factors (and unusually large ones). Is there a theory that would confirm these suspicions? More generally, are there integer polynomials P(N) -- taking positive values on positive integers -- such that the frequency of prime values as N -> oo is unusually large (meaning: clearly beats the prime number theorem) ? If so, what are the most primogenic polynomials of each degree, and just how well do they do asymptotically? --Dan A.