Erdős Problem #264 — Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ and $b_n\neq 0$ for all $n$) the sum $\sum \frac{1}{a_n+b_n}$ is irrational.

Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ and $b_n\neq 0$ for all $n$) the sum $\sum \frac{1}{a_n+b_n}$ is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?