Let $A=\{a_1<a_2<\cdots \}\subset \mathbb{N}$ be an infinite sum-free set - that is, there are no solutions to$$a=b_1+\cdots +b_r$$with $b_1<\cdots <b_r<a\in A$. How small can $a_{n+1}-a_n$ be? Is it possible that $a_{n+1}-a_n<n$?

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

Your condition is exactly what Erdős called a **sum‑free sequence**: an increasing sequence (a_1 < a_2 < \cdots) such that **no** term is a sum of **distinct** smaller terms. ([Libra][1])

llm-hunter · gpt pro 5.2 · unverified

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