Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\ne 0$ and $b_n\ne 0$ for all $n$) 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?

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

This property is studied in the literature under the name **Type 3 irrationality sequence** (Erdős–Graham): an increasing integer sequence ((a_n)) such that for **every bounded** integer sequence ((b_n)) with (b_n\neq 0) and (a_n+b_n\neq 0) for all $n$, the “perturbed” Ahmes series [ \sum_{n=1}^\infty \frac{1}{a_n+b_n}…

llm-hunter · gpt pro 5.2 · unverified

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