{"schema":"vela.problem-packet.v0.1","problem":1204,"statement":"We call a sequence of integers $0\\leq a_1<\\cdots <a_k$ admissible if it is missing at least one congruence class modulo every prime $p$. Let $A(k)=\\min a_k$. Estimate $A(k)$ - in particular, is it true that\\[A(k)\\sim k\\log k?\\]Estimate\\[B(k)=\\min \\frac{a_1+\\cdots+a_k}{k}.\\]","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A008407","name":"Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations.","terms":"0,2,6,8,12,16,20,26,30,32,36,42,48,50,56,60,66,70,76,80,84,90,94,100,110,114,120,126,130,136,140,146,152,156,158,162,168","url":"https://oeis.org/A008407"},{"id":"A023193","name":"a(n) gives the largest number k for which there is at least one admissible k-tuple taken from [0, 1, ..., n-1] if the tuple starts with 0. Admissibility is defined in a comment.","terms":"1,1,2,2,2,2,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,12,13,13,14","url":"https://oeis.org/A023193"},{"id":"A135311","name":"A greedy sequence of prime offsets.","terms":"0,2,6,8,12,18,20,26,30,32,36,42,48,50,56,62,68,72,78,86,90,96,98,102,110,116,120,128,132,138,140,146,152,156,158,162,168","url":"https://oeis.org/A135311"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}