OPEN
Let $a_n$ be a sequence such that $a_n/n\to \infty$. Is the sum
\[\sum_n \frac{a_n}{2^{a_n}}\]
irrational?
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.
