{"schema":"vela.problem-packet.v0.1","problem":342,"statement":"With $a_1=1$ and $a_2=2$ let $a_{n+1}$ for $n\\geq 2$ be the least integer $>a_n$ which can be expressed uniquely as $a_i+a_j$ for $i<j\\leq n$.What can be said about this sequence? Do infinitely many pairs $a,a+2$ occur? Does this sequence eventually have periodic differences? Is the density $0$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_2c11f36ee072ef95","kind":"dead_end","claim":"attempted via frontier 'sidon/B2' (transfer_strength=none) -> no_progress","grade":"honest_null","gateStatus":"needs_verification","superseded":false}],"velaLean":[],"oeis":[{"id":"A002858","name":"Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.","terms":"1,2,3,4,6,8,11,13,16,18,26,28,36,38,47,48,53,57,62,69,72,77,82,87,97,99,102,106,114,126,131,138,145,148,155,175,177,180,","url":"https://oeis.org/A002858"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}