Let $t_k(n)$ denote the least $m$ such that$$n\mid m(m+1)(m+2)\cdots (m+k-1).$$Is it true that$$\sum _{n\le x}{t}_2(n)\ll \frac{x^2}{(\log x{)}^c}$$for some $c>0$?Is it true that, for $k\ge 2$,$$\sum _{n\le x}{t}_{k+1}(n)=o\left(\sum _{n\le x}{t}_k(n)\right)?$$

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

These are classical open questions of Erdős–Hall (1978) / Erdős–Graham (1980). They appear as Erdős Problem #394 and are still listed as open. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

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