{"schema":"vela.problem-packet.v0.1","problem":1108,"statement":"Let\\[A = \\left\\{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\textrm{ finite}\\right\\}.\\]If $k\\geq 2$, then does $A$ contain only finitely many $k$th powers? Does it contain only finitely many powerful numbers?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A025494","name":"Squares which are the sum of factorials of distinct integers (probably finite).","terms":"1,4,9,25,121,144,729,841,5041,5184,45369,46225,363609,403225,3674889,1401602635449","url":"https://oeis.org/A025494"},{"id":"A051761","name":"Numbers that are simultaneously a sum of factorials of distinct integers and of the form a^b with b >= 2.","terms":"0,1,4,8,9,25,27,32,121,128,144,729,841,5041,5184,45369,46225,363609,403225,3674889,1401602635449","url":"https://oeis.org/A051761"},{"id":"A115645","name":"Powerful(1) numbers (A001694) that are sums of distinct factorials.","terms":"1,8,9,25,27,32,121,128,144,729,841,864,5041,5184,40328,41067,45369,45387,46208,46225,363609,403225,3674889,43954688,6230","url":"https://oeis.org/A115645"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}