Product representations of perfect powers

Let $ρ_k(N)$ 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 $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right) +O_d(π(N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right)$. This answers a question of Verstraëte.