Problem 247

Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that

\limsup \frac{a_n}{n}=\infty.

Is

\sum_{n=1}^\infty \frac{1}{2^{a_n}}

transcendental?

Status

Open

No claim settles this problem.

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Er57: Erdős, P., On the irrationality of certain series. Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 (1957), 212--219, pp. 213, 215.