Let $g(n)$ be the maximal size of $A\subseteq \{1,\dots ,n\}$ such that the products ${\prod }_{n\in S}n$ are distinct for all $S\subseteq A$. Is it true that$$g(n)\le \pi (n)+\pi (n^{1/2})+o\left(\frac{x^{1/2}}{\log n}\right)?$$