Erdős Problem #263

We call a **strictly increasing** sequence \(a_n\) of positive integers an _irrationality sequence_ if for any sequence \(b_n\) of positive integers with \(\frac{a_n}{b_n} \to 1\) as \(n \to \infty\), the sum \(\sum \frac{1}{b_n}\) converges to an irrational number. Note: erdosproblems.com/263 was corrected on 2026-04-02 to require the sequence to be increasing; the pre-correction statement (no monotonicity hypothesis) had a counterexample to Q2 that is not increasing (see `erdos_263.parts.ii` below). Not

The nondecreasing version of `IsIrrationalitySequence`, as used by Koizumi [Ko25]: the sequence is positive and nondecreasing, and every positive sequence asymptotic to it has irrational reciprocal sum.