Let $S$ denote the set of natural numbers $m$ with the property that for all prime powers $p^k || m$ we have $k \equiv 1 \pmod{2}$.

What is the asymptotic density of $S$?

Note that $S$ contains all prime numbers and more generally, all square-free numbers, so that $\liminf_{X \rightarrow \infty} \frac{\# (S \cap [1,X])}{X} \geq \frac{6}{\pi^2}$.