{"schema":"vela.problem-packet.v0.1","problem":359,"statement":"Let $a_1<a_2<\\cdots$ be an infinite sequence of integers such that $a_1=n$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. What can be said about the density of this sequence?In particular, in the case $n=1$, can one prove that $a_k/k\\to \\infty$ and $a_k/k^{1+c}\\to 0$ for any $c>0$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_55ec76001c9c7795","kind":"dead_end","claim":"attempted via frontier '?' (transfer_strength=n/a) -> no_progress","grade":"honest_null","gateStatus":"needs_verification","superseded":false}],"velaLean":[],"oeis":[{"id":"A002048","name":"Segmented numbers, or prime numbers of measurement.","terms":"1,2,4,5,8,10,14,15,16,21,22,25,26,28,33,34,35,36,38,40,42,46,48,49,50,53,57,60,62,64,65,70,77,80,81,83,85,86,90,91,92,10","url":"https://oeis.org/A002048"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}