{"schema":"vela.problem-packet.v0.1","problem":945,"statement":"Let $F(x)$ be the maximal $k$ such that there exist $n+1,\\ldots,n+k\\leq x$ with $\\tau(n+1),\\ldots,\\tau(n+k)$ all distinct (where $\\tau(m)$ counts the divisors of $m$). Estimate $F(x)$. In particular, is it true that\\[F(x) \\leq (\\log x)^{O(1)}?\\]In other words, is there a constant $C>0$ such that, for all large $x$, every interval $[x,x+(\\log x)^C]$ contains two integers with the same number of divisors?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A048892","name":"Start of n consecutive integers with distinct number of divisors.","terms":"1,1,4,9,45,76,270,2204,3718,95499,590890,16023339,16475964,1745175039,31287652672,347321438520,2620400333120,23991979183","url":"https://oeis.org/A048892"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}