{"schema":"vela.problem-packet.v0.1","problem":852,"statement":"Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n<x$ the numbers $d_n,d_{n+1},\\ldots,d_{n+h(x)-1}$ are all distinct. Estimate $h(x)$. In particular, is it true that\\[h(x) >(\\log x)^c\\]for some constant $c&#62;0$, and\\[h(x)=o(\\log x)?\\]","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A001223","name":"Prime gaps: differences between consecutive primes.","terms":"1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6,6,4,6,6,2,10,2,4,2,12,12,4,2,4,6,2,10,6,6,6,2,","url":"https://oeis.org/A001223"},{"id":"A053597","name":"Let prime(i) = i-th prime (A000040), let d(i) = prime(i+1)-prime(i) (A001223); a(n) = number of distinct numbers among d(n), d(n+1), d(n+2), ... before first duplicate is encountered.","terms":"2,1,2,2,2,2,3,3,2,3,3,2,3,2,1,2,3,3,3,3,2,3,4,3,2,2,2,3,2,5,4,3,2,3,2,1,2,2,1,3,2,3,2,3,2,1,3,2,3,4,3,3,2,1,1,2,3,5,4,4,","url":"https://oeis.org/A053597"},{"id":"A078515","name":"Numbers n such that A053597(n) sets a new record.","terms":"1,7,23,30,94,219,279,773,1856,3724,6999,7000,19205,184163,280103,849876,1870722,3570761,4114341,11271072,55282774,682560","url":"https://oeis.org/A078515"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}