Let $1\le a_1<a_2<\cdots$ be a sequence of integers such that no $a_i$ is the sum of consecutive $a_j$ for $j<i$. Is it true that$$\mathrm{limsup}\frac{a_n}{n}=\infty ?$$Or even$$\lim \frac{1}{\log x}\sum _{a_n<x}\frac{1}{a_n}=0?$$

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

This is (to my knowledge) still **open** in the form you stated it: it is listed as Erdős problem **#839** (“no term is the sum of a consecutive block of earlier terms”), and neither of your two limits is presently known to hold in full generality. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

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