Let $A(n)=\{a_0<a_1<\cdots \}$ be the sequence defined by $a_0=0$ and $a_1=n$, and for $k\ge 1$ define $a_{k+1}$ as the least positive integer such that there is no three-term arithmetic progression in $\{a_0,\dots ,a_{k+1}\}$.Can the $a_k$ be explicitly determined? How fast do they grow?

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

What you’ve defined is exactly the **Stanley sequence** generated from the 3‑free seed set $\\{0,n\\}$. In the usual notation it’s (S(0,n)={a_k}_{k\ge0}), built by the greedy rule “take the smallest integer (>a_k) that keeps the set 3‑term‑AP‑free.” ([MIT Mathematics][1])

llm-hunter · gpt pro 5.2 · unverified

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