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Let $a_n$ be a sequence of integers such that, for every bounded sequence $b_n$, 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? Is there such a sequence with $a_n<n^k$?
A possible definition of an 'irrationality sequence' (see also [262] and [263]). One example is $a_n=2^{2^n}$.