{"schema":"vela.problem-packet.v0.1","problem":679,"statement":"Let $\\epsilon>0$ and $\\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that\\[\\omega(n-k) < (1+\\epsilon)\\frac{\\log k}{\\log\\log k}\\]for all $k<n$ which are sufficiently large depending on $\\epsilon$ only?Can one show the stronger version with\\[\\omega(n-k) < \\frac{\\log k}{\\log\\log k}+O(1)\\]is false?","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)"}