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 an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. Does there exist some absolute constant $c > 0$ such that there are always infinitely many $N$ with $|A \cap \{1, \dotsc, N\}| < N^{1−c}$?

The DeepMind prover agent has found a formal disproof of this statement.