{"schema":"vela.problem-packet.v0.1","problem":345,"statement":"Let $A\\subseteq \\mathbb{N}$ be a complete sequence, and define the threshold of completeness $T(A)$ to be the least integer $m$ such that all $n\\geq m$ are in\\[P(A) = \\left\\{\\sum_{n\\in B}n : B\\subseteq A\\textrm{ finite }\\right\\}\\](the existence of $T(A)$ is guaranteed by completeness). Is it true that there are infinitely many $k$ such that $T(n^k)>T(n^{k+1})$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A001661","name":"Largest number not the sum of distinct positive n-th powers.","terms":"128,12758,5134240,67898771,11146309947,766834015734,4968618780985762","url":"https://oeis.org/A001661"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}