Problem 264

Erdős Problem #264: declared status 'open'. Formalized: yes. Let $a_n$ be a sequence of 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? Prize: no. Tags: irrationality.