Erdős Problem #263 — Let $a_n$ be a sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\to 1$ the sum $\sum\frac{1}{b_n}$ is irrational. Is $a_n=2^{2^n}$ such a sequence?

Let $a_n$ be a sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\to 1$ the sum $\sum\frac{1}{b_n}$ is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to \infty$?