Erdős Problem #260

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?

#260: [Er74b][ErGr80][Er81h,p.180][Er88c,p.103]

irrationality

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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.

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This page was last edited 19 October 2025.

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