Is there a sequence $A=\{a_1\le a_2\le \cdots \}$ of integers with$$\lim \frac{a_{n+1}}{a_n}=2$$such that$$P(A^{\prime })=\left\{\sum _{n\in B}n:B\subseteq A^{\prime }\text{ finite }\right\}$$has density $1$ for every cofinite subsequence $A^{\prime }$ of $A$?

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

As of **January 16, 2026**, this is **not known**: the existence of such a sequence $A$ with [ \lim_{n\to\infty}\frac{a_{n+1}}{a_n}=2 ] and with the property that **for every cofinite subsequence** (A'\subseteq A), the finite subset-sum set [ P(A')=\\{\sum_{n\in B} n : B\subseteq A'\ \text{finite}\\} ] has **(natural) …

llm-hunter · gpt pro 5.2 · unverified

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