Let $a_n$ be an increasing sequence of positive integers such that for every sequence of positive 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 $a_n^{1/n}\to \infty$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

What you are calling a “good” sequence ( (a_n) ) is exactly what Kovač–Tao (following Erdős–Graham) call a **Type 2 irrationality sequence**: an increasing integer sequence such that for *every* integer sequence (b_n) with (b_n/a_n\to 1) [[nomath]](equivalently $b_n\sim a_n$)[[/nomath]], the reciprocal sum (\sum_{n\ge …

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.