{"schema":"vela.problem-packet.v0.1","problem":218,"statement":"Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A064113","name":"Indices k such that (1/3)*(prime(k)+prime(k+1)+prime(k+2)) is a prime.","terms":"2,15,36,39,46,54,55,73,102,107,110,118,129,160,164,184,187,194,199,218,239,271,272,291,339,358,387,419,426,464,465,508,5","url":"https://oeis.org/A064113"},{"id":"A333230","name":"Positions of weak ascents in the sequence of differences between primes.","terms":"1,2,3,5,7,8,10,13,14,15,17,20,22,23,26,28,29,31,33,35,36,38,39,41,43,45,46,49,50,52,54,55,57,60,61,64,65,67,69,70,71,73,","url":"https://oeis.org/A333230"},{"id":"A333231","name":"Positions of weak descents in the sequence of differences between primes.","terms":"2,4,6,9,11,12,15,16,18,19,21,24,25,27,30,32,34,36,37,39,40,42,44,46,47,48,51,53,54,55,56,58,59,62,63,66,68,72,73,74,77,8","url":"https://oeis.org/A333231"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}