{"schema":"vela.problem-packet.v0.1","problem":892,"statement":"Is there a necessary and sufficient condition for a sequence of integers $b_1<b_2<\\cdots$ that ensures there exists a primitive sequence $a_1<a_2<\\cdots$ (i.e. no element divides another) with $a_n \\ll b_n$ for all $n$?In particular, is this always possible if there are no non-trivial solutions to $(b_i,b_j)=b_k$?Similarly, find necessary and sufficient conditions on a sequence $n_1<n_2<\\cdots$ that ensure there exists a primitive set $A$ such that\\[\\lvert A\\cap [1,2^{n_i}]\\rvert \\gg 2^{n_i}\\]for every $i$.","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)"}