Problem Statement
Let $a_1<a_2<\cdots$ be an increasing sequence such that $a_n/n\to \infty$. Is the sum\[\sum_n \frac{a_n}{2^{a_n}}\]irrational?
Categories:
Irrationality
Progress
Erdős [Er81l] proved this is true under either of the stronger assumptions that- $a_{n+1}-a_n\to \infty$ or
- $a_n \gg n\sqrt{\log n\log\log n}$.
Erdős and Graham speculate that the condition $\limsup a_{n+1}-a_n=\infty$ is not sufficient, but know of no example.
Source: erdosproblems.com/260 | Last verified: January 14, 2026