Let $A = {a_{1} < a_{2} < \dots}$ be an infinite sequence of positive integers with positive lower logarithmic density.Must there exist a sequence $n_{1} < \dots$ such that $a_{n_{i}} ∣ a_{n_{i + 1}}$ for all $i \geq 1$ and $x \to \infty lim sup \frac{1}{lo g lo g x} a_{n_{i}} < x \sum 1 \geq x \to \infty lim sup \frac{1}{lo g lo g x} a_{n} < x \sum \frac{1}{a _{n} lo g a _{n}} ?$

Worked, still open.

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