Problem 1101

Erdős Problem #1101: declared status 'open'. Formalized: yes. If $u=\{u_1<u_2<\cdots\}$ is a sequence of integers such that $(u_i,u_j)=1$ for all $i\neq j$ 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$$u_1\cdots u_{t_x}\leq x< u_1\cdots u_{t_x}u_{t_x+1}.$$We call such a sequence $u_i$ good if, for all $\epsilon>0$, if $x$ is sufficiently large then$$\max_{a_k<x} (a_{k+1}-a_k) < (1+\epsilon)t_x \prod_{i}\left(1-\frac{1}{u_i}\right)^{-1}.$$Is there a good sequence such that $u_n< n^{O(1)}$? Is there a good sequence such that $u_n\leq e^{o(n)}$? Prize: no. Tags: number theory.