Given a polynomial $f(x) \in \mathbb{Z}[x]$ of degree $d$, consider the following three sets:

$$N_1(x) = \#\{k \leq x: f(k) \text{ is square-free}\}$$ $$N_2(x) = \#\{n \leq x: n = f(k) \text{ is square-free}\}$$ $$N_3(x) = \#\{n \leq x: n \text{ is the square-free part of $f(k)$}\}$$

Generally, we should have that $N_1(x) \sim c x$ (not proven for large degrees, assumes that $f$ satisfies the necessary congruence conditions). Then $N_2(x) \sim c' x^{1/d}$. What do we know about the growth of $N_3(x)$? Do we get significantly more values by considering square-free parts and not just square-free values?

Any references for this would be appreciated.

Edit: Changed from big-O notation to make order of growth more precise.

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