Reading the following link : https://en.m.wikipedia.org/wiki/Bateman-Horn_conjecture I realized that the quantity $ N(p) $ doesn't depend so much upon $ n $ than upon its radical. So let's say that two integers $ m $ and $ n $ below $ x $ are $ x $-conjugates if and only if $ rad(m)=rad(n) $ . So that the quantity $ N(p) $ should be preserved under the action of the group of permutations of $ x $ -conjugates integers. Among those permutations, some should be field automorphisms and thus preserve the expression of the quantity $ P(x) $.
So can considering this group shed some light on the Bateman-Horn conjecture ?