OPENHARDnumber theory
Let $A$ be an infinite set with no distinct $a,b,c \in A$ where $a \mid (b+c)$ and $b,c>a$. Does such an $A$ exist with $\liminf |A \cap \{1,\ldots,N\}| / N^{1/2} > 0$? Must $\sum_{n \in A} 1/n < \infty$?
Notes: Erdos-Sarkozy proved such A must have density 0. Elsholtz-Planitzer constructed sets with |A cap {1,...,N}| >> N^{1/2} / (log N)^{1/2} (log log N)^2 (log log log N)^2.
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