OPEN
Let $a_n$ be a sequence of integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ for all $n$ and $b_n$ is not identically zero) the sum
\[\sum \frac{1}{a_n+b_n}\]
is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?
A possible definition of an 'irrationality sequence' (see also
[262] and
[263]). One example is $a_n=2^{2^n}$. In
[ErGr80] they also ask whether such a sequence can have polynomial growth, but Erdős later retracted this in
[Er88c], claiming 'It is not hard to show that it cannot increase slower than exponentially'.
Kovač [Ko24c] has proved that $2^n$ is not such an irrationality sequence. More generally, he proves that any strictly increasing sequence of positive integers such that $\sum\frac{1}{a_n}$ converges and for which there exists $C>1$ such that $\frac{1}{a_n^2}\leq C\sum_{k>n}\frac{1}{a_k^2}$ is not such an irrationality sequence.
