OPEN
Let $a_n$ be a sequence of integers such that for every sequence of integers $b_n$ with $b_n/a_n\to 1$ the sum
\[\sum\frac{1}{b_n}\]
is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to \infty$?
One possible definition of an 'irrationality sequence' (see also
[262] and
[264]). A folklore result states that $\sum \frac{1}{a_n}$ is irrational whenever $\lim a_n^{1/2^n}=\infty$.
Kovač and Tao [KoTa24] have proved that any strictly increasing sequence such that $\sum \frac{1}{a_n}$ converges and $\lim a_{n+1}/a_n^2=0$ is not such an irrationality sequence. On the other hand, if
\[\liminf \frac{a_{n+1}}{a_n^{2+\epsilon}}>0\]
for some $\epsilon>0$ then the above folklore result implies that $a_n$ is such an irrationality sequence.