Problem 263

Erdős Problem #263: declared status 'open'. Formalized: yes. Let $a_n$ be a sequence of integers such that for every sequence of 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 $a_n^{1/n}\to \infty$? Prize: no. Tags: irrationality.