{"schema":"vela.problem-packet.v0.1","problem":1101,"statement":"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&#60;x} (a_{k+1}-a_k) &#60; (1+\\epsilon)t_x \\prod_{i}\\left(1-\\frac{1}{u_i}\\right)^{-1}.\\]Is there a good sequence such that $u_n&#60; n^{O(1)}$? Is there a good sequence such that $u_n\\leq e^{o(n)}$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}