{"schema":"vela.problem-packet.v0.1","problem":420,"statement":"If $\\tau(n)$ counts the number of divisors of $n$ then let\\[F(f,n)=\\frac{\\tau((n+\\lfloor f(n)\\rfloor)!)}{\\tau(n!)}.\\]Is it true that\\[\\lim_{n\\to \\infty}F((\\log n)^C,n)=\\infty\\]for large $C$? Is it true that $F(\\log n,n)$ is everywhere dense in $(1,\\infty)$? More generally, if $f(n)\\leq \\log n$ is a monotonic function such that $f(n)\\to \\infty$ as $n\\to \\infty$, then is $F(f,n)$ everywhere dense?","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)"}