If $u=\{u_1<u_2<\cdots \}$ is a sequence of integers such that $(u_i,u_j)=1$ for all $i\ne j$ and $\sum \frac{1}{u_i}<\infty$ then let $\{a_1<a_2<\cdots \}$ be the sequence of integers which are not divisible by any of the $u_i$. For any $x$ define $t_x$ by$$u_1\cdots u_{t_x}\le x<u_1\cdots u_{t_x}{u}_{t_x+1}.$$We call such a sequence $u_i$ good if, for all $ϵ>0$, if $x$ is sufficiently large then$$\underset{a_k<x}{\max }(a_{k+1}-a_k)<(1+ϵ)t_x\prod _i{\left(1-\frac{1}{u_i}\right)}^{-1}.$$Is there a good sequence such that $u_n<n^{O(1)}$? Is there a good sequence such that $u_n\le e^{o(n)}$?

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

This is essentially Erdős’s problem about **gaps between (\mathcal B)-free numbers** [[nomath]](here $\mathcal B={u_i}$)[[/nomath]]: the set $ \\{a_1<a_2<\cdots\\}=\\{n\ge1:\ \forall i,\ u_i\nmid n\\} $ has natural density [ \delta:=\prod_{i}\\(1-\frac1{u_i}\\)\in(0,1], ] and the question is whether one can make the **…

llm-hunter · gpt pro 5.2 · unverified

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