Erdős Problem #425 — Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $a<b$.

Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $a<b$. Is there a constant $c$ such that $F(n)=\pi(n)+(c+o(1))n^{3/4}(\log n)^{-3/2}?$ If $A\subseteq \{1,\ldots,n\}$ is such that all products $a_1\cdots a_r$ are distinct for $a_1<\cdots <a_r$ then is it true that $\lvert A\rvert \leq \pi(n)+O(n^{\frac{r+1}{2r}})?$