{"schema":"vela.problem-packet.v0.1","problem":1100,"statement":"If $1=d_1<\\cdots<d_{\\tau(n)}=n$ are the divisors of $n$, then let $\\tau_\\perp(n)$ count the number of $i$ for which $(d_i,d_{i+1})=1$.Is it true that $\\tau_\\perp(n)/\\omega(n)\\to \\infty$ for almost all $n$? Is it true that\\[\\tau_\\perp(n)< \\exp((\\log n)^{o(1)})\\]for all $n$?Let\\[g(k) = \\max_{\\omega(n)=k}\\tau_\\perp(n),\\]where $\\omega(n)$ counts the number of distinct prime divisors of $n$, and $n$ is restricted to squarefree integers. Determine the growth of $g(k)$.","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A325864","name":"Number of subsets of {1..n} of which every subset has a different sum.","terms":"1,2,4,7,13,22,36,56,91,135,211,307,446,625,882,1194,1677,2238,3031,4001,5460,6995,9302,11921,15424,19554,25032,31005,391","url":"https://oeis.org/A325864"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}