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Let $A\subseteq \mathbb{N}$ be an infinite set. Is \[\sum_{n\in A}\frac{1}{2^n-1}\] irrational?
If $A=\mathbb{N}$ then this series is $\sum_{n}\frac{d(n)}{2^n}$, where $d(n)$ is the number of divisors of $n$, which Erdős [Er48] proved is irrational.