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?

An example of an $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ and such that $$\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\log N > 0$$ is given by the set of $p^2$, where $p\equiv 3\pmod{4}$ is prime.