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Claim. P. Erdős, On the irrationality of certain series, Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 (1957), no. 2, 212--219, communicated at the Academy's meeting of 29 December 1956. Theorem 2 (printed p. 215; proof pp. 218--219) states: let $t>1$ be an integer and let $1<n_1<n_2<\cdots$ be integers with $\limsup_{k\to\infty}n_k/k^l=\infty$ for a positive integer $l$; then

$$\sum_{k=1}^{\infty}\frac{1}{t^{n_k}}$$

satisfies no algebraic equation with integer coefficients of degree at most $l$. The result page Theorem 2 of the source card erdos_1957_irrationality_certain_series gives the statement and the structure of the proof, which expands a supposed integer equation by the multinomial theorem and applies the paper's Lemma 4, an irrationality criterion for sparse series with signed integer coefficients. On p. 213 the paper recalls that Erdős and Straus had shown $\sum_k1/t^{n_k}$ transcendental when $\limsup\log n_k/\log k=\infty$, citing Elem. Math. 9 (1954), p. 18, Problem 154, says that Theorem 2 comes from a modification of that method, and then asks the question of Problem 247 for a general base $t$: whether a series with $\limsup n_k/k=\infty$ can be algebraic, adding that even for $n_k>ck^2$ it is not known whether the square of the sum is irrational.

Covers. The series $\sum_n2^{-a_n}$ is transcendental for every increasing sequence of positive integers with $\limsup a_n/n^t=\infty$ for every $t\ge1$, the statement the site credits; the condition is equivalent to $\limsup\log a_n/\log n=\infty$. If the sum were algebraic of degree $d$, Theorem 2 with $t=2$ and $l=d$, applied to the terms $a_n>1$ (a first term $a_1=1$ contributes $1/2$, which changes neither algebraicity nor degree), would be contradicted, since $\limsup a_n/n^d=\infty$. Under the problem's own hypothesis $\limsup a_n/n=\infty$, Theorem 2 gives irrationality only ($l=1$), which settles no instance of the question; the sequences that beat $n$ but not every power of $n$ are the open part.

Acceptance. Refereed: the journal Indagationes Mathematicae, the mathematical series of the Proceedings of the Royal Netherlands Academy, volume 19 (1957), communicated by J. Popken. The site labels the problem OPEN and credits the statement to [Er75c], a 1975 paper that does not contain it (see the problem page); its remark on an open problem is not reviewed evidence. The proof is not checked here.

Depends on. Nothing in this wiki.