Product representations of perfect powers

Let ${\rho }_k(N)$ denote the maximum size of a set $A\subseteq \{1,2,\dots ,N\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that ${\rho }_d(N)={\sum }_{k=1}^{d-1}\pi \left(\frac{N}{k}\right)+O_d(\pi (N^{1\mathrm{/}2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have ${\rho }_d(N)={\sum }_{k=1}^{d-1}\pi \left(\frac{N}{k}\right)$. This answers a question of Verstraëte.