Let $S$ be the set of positive integers $n$ with the property that exactly one prime factor of $n$ has multiplicity $1$ and every other prime factor has multiplicity greater than $1$ (to be clear, $S$ does not contain the prime numbers). What is the asymptotic behavior of $\frac{\#\{n \in S' : n \leq X\}}{\#\{n \leq X\}}$ in the limit as $X \to \infty$?

What I know: There is an easier analogue of this problem that I have an approach for, namely finding the density of the set $S'$ of positive integers $n$ such that every prime factor of $n$ has multiplicity greater than $1$. It is not hard to check that $$\frac{\#\{n \in S' : n \leq X\}}{\#\{n \leq X\}} \leq \int_{1}^X (X/n^2)^{1/3} dn \approx 3 \sqrt{X}.$$ I suspect that the density of $S$ must be in the literature somewhere, but I can at least use the density of $S'$ to get an upper bound on the density of $S$ as follows: $$\frac{\#\{n \in S : n \leq X\}}{\#\{n \leq X\}} \leq \sum_{p \text{ prime}} \frac{\#\{n \in S' : n \leq X/p\}}{\#\{n \leq X/p\}} \leq \sum_{p \text{ prime}}3 \sqrt{X/p} \ll X/\log X,$$ where in the last step above, I've used a well-known estimate for the sum of the reciprocals of the square roots of the prime numbers. Can one do better than $X/\log X$? (Is a power savings possible?)