Recall that an integer $n$ is called *squareful* if for every prime $p$ with $p \mid n$, we also have $p^2 \mid n$.


Any squareful number can be written uniquely as $n= x^2 y^3$ where $y$ is squarefree. From this, it is easy to see that
$$\#\{ n \in \mathbb{Z}: |n| \leq X, \, n \text{ is squareful} \} \ll X^{1/2}.$$


I would like a version of this for polynomials.



> 
> Let $f \in \mathbb{Z}[x]$ be non-constant and separable. Then does there exist $\delta > 0$ such that
>  $$\#\{ n \in \mathbb{Z}: |n| \leq X, \, f(n) \text{ is squareful} \} \ll X^{1 - \delta} \quad ?$$
> 
> 
> 


Hopefully there are not some necessary local conditions here that I overlooked. In my application I am happy to change $f$ as required so that one can assume that $f$ is sufficiently "general". Moreover, I can even assume that $f$ is of very large degree if necessary to simplify things.


I would normally try to prove something like this using the large sieve, however the large sieve gives the poor upper bound $X/(\log X)$, whereas I would like a power saving.


If necessary, I'm happy to assume some standard conjectures (e.g. the abc conjecture).



Squareful values of polynomials
If $f$ is squarefree and of degree at least three, then
the abc conjecture implies $f(x)$ is squareful finite number
of times. For [reference](http://www.math.unicaen.fr/~nitaj/abc.html#Consequences) and several papers of Granville:



> 
> Power free-values of polynomials. Langevin noted in [Lan2] the following conjecture which is a consequence of the abc conjecture.
>  Let $F(X)$ be a polynomial with integer coefficients and no repeated roots. For any $\epsilon > 0$, there exists a constant $C\_{\epsilon,F}$ such that for any integer $n$,$|n|^{\deg(F)-1-\epsilon} < C\_{\epsilon,F} rad(F(n))$.
> 
> 
> 


Actually abc implies the squareful part of $f(x)$ can't be too large
infinitely often. If the squareful part of $f(x)$ is $s$, then
$ s < C\_\epsilon x^{2+\epsilon}$.


