{"schema":"vela.problem-packet.v0.1","problem":983,"statement":"Let $n\\geq 2$ and $\\pi(n)<k\\leq n$. Let $f(k,n)$ be the smallest integer $r$ such that in any $A\\subseteq \\{1,\\ldots,n\\}$ of size $\\lvert A\\rvert=k$ there exist primes $p_1,\\ldots,p_r$ such that $>r$ many $a\\in A$ are only divisible by primes from $\\{p_1,\\ldots,p_r\\}$. Is it true that\\[2\\pi(n^{1/2})-f(\\pi(n)+1,n)\\to \\infty\\]as $n\\to \\infty$?In general, estimate $f(k,n)$, particularly when $\\pi(n)+1&#60;k=o(n)$.","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)"}