Erdős Problem #327 — Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ and $a\neq b$ then $a+b mid ab$. Can $A$ be 'substantially more' than the odd numbers?

Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ and $a\neq b$ then $a+b mid ab$. Can $A$ be 'substantially more' than the odd numbers? What if $a,b\in A$ with $a\neq b$ implies $a+b mid 2ab$? Must $\lvert A\rvert=o(N)$?