Let $A \subset \mathbb{N}$ be infinite with no distinct $a, b, c \in A$ such that $a \mid (b + c)$ with $b, c > a$. Can $|A \cap [1, N]|/\sqrt{N}$ have positive lower limit? Must every such $A$ fall below $N^{1-c}$ infinitely often?

Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then $$\lvert A\cap\{1,\ldots,N\}\rvert \ll N^{2/3}/\log N$$