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Claim. Theorem 3 (p. 6) of P. Erdős, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 (1975), 1--7, states that if $n_k>0$ and $n_k\equiv0\pmod{2^{2^k}}$ for every $k$, then

$$\sum_{k\ge1}\frac{1}{n_k}$$

is irrational; the paper notes that the sequence $n_k$ is not assumed monotone. On pp. 2--3 Erdős recalls the property $P$ he and Straus considered, that a sequence $n_1<n_2<\cdots$ has property $P$ when $\sum_k1/m_k$ is irrational for every choice of positive integers $m_k$ divisible by $n_k$, which is the notion of an irrationality sequence in Problem 262 with $m_k=t_ka_k$; he writes that they wondered whether $n_k=2^{2^k}$ has property $P$ and that he will prove this conjecture, which Theorem 3 does. So $a_n=2^{2^n}$ is an irrationality sequence, with $\limsup(\log_2\log_2a_n)/n=1$. The proof reorders the $m_k$ into a monotone sequence, reduces by Theorem 1 to the case $\limsup m_k^{1/2^k}<\infty$, and uses that at least two of $m_1,\ldots,m_k$ are divisible by $2^{2^{k-1}}$, so the least common multiple $N_k$ of $m_1,\ldots,m_k$ is at most $2^{-2^{k-1}}$ times their product; along a subsequence where $m_k$ is nearly as large as the growth allows, the product times the tail tends to $0$, which rationality forbids. The paper adds that property $P$ is of interest only when $\lim n_k^{1/2^k}<\infty$, and that Erdős did not know whether a sequence with property $P$ can grow slowly, the problem's question. The source card is erdos_1976_problems_results_irrationality_sum_infinite_series, and the result page is Theorem 3.

Covers. The attained half of the answer: an irrationality sequence can grow as slowly as $2^{2^n}$, so $\limsup(\log_2\log_2a_n)/n=1$ is attained. Not covered: the lower bound, that no irrationality sequence has $\limsup(\log_2\log_2a_n)/n<1$, which is Hančl's theorem on its claim page.

Acceptance. Refereed: the Journal of Mathematical Sciences, volume 10 (1975), pp. 1--7. The site's remarks credit this paper with the example, but the SOLVED (LEAN) label credits Hančl with the answer, so no reviewed evidence is listed. The proof is not checked here.

Depends on. Nothing in this wiki.