Let $n \in \mathbb{N}$. Define the *radical* $R(n)$ of $n$ by

$$\displaystyle R(n) = \prod_{p | n} p.$$

In other words, $R(n)$ is the largest square-free number which divides $n$.

For an integer $k \geq 2$ we say that an integer $n$ is $k$-free (generalization of square-free) if $p | n$ implies that $p^k \nmid n$.

Define

$$\displaystyle S_k(X) = \# \{n \in \mathbb{N} : n \text{ is } k\text{-free}, R(n) \leq X\}.$$

When $k = 2$ this is just the count for square-free numbers up to $X$, and it is well-known that $S_2(X) = \frac{6}{\pi^2} X + O(\sqrt{X})$.

How does one obtain the asymptotic expression for $S_k(X)$ (if one exists) for $k \geq 3$?