{"schema":"vela.problem-packet.v0.1","problem":283,"statement":"Let $p:\\mathbb{Z}\\to \\mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\\geq 2$ with $d\\mid p(n)$ for all $n\\geq 1$. Is it true that, for all sufficiently large $m$, there exist integers $1\\leq n_1<\\cdots <n_k$ such that\\[1=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k}\\]and\\[m=p(n_1)+\\cdots+p(n_k)?\\]","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A380791","name":"For a positive rational x, let k(x) be the smallest positive integer such that all k >= k(x) have a partition into distinct parts with reciprocal sum equal to x. The n-th term in this sequence is equa","terms":"2,2,2,1,2,4,5,5,7,7,5,12,18,22,32,38,41,48,57,76,82,74,97,117,155,170,194,228,277,306,332,430,473,483,510","url":"https://oeis.org/A380791"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}