{"schema":"vela.problem-packet.v0.1","problem":935,"statement":"For any integer $n=\\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that\\[Q_2(n) = \\prod_{\\substack{p\\\\ k_p\\geq 2}}p^{k_p}.\\]Is it true that, for every $\\epsilon>0$ and $\\ell\\geq 1$, if $n$ is sufficiently large then\\[Q_2(n(n+1)\\cdots(n+\\ell))<n^{2+\\epsilon}?\\]If $\\ell\\geq 2$ then is\\[\\limsup_{n\\to \\infty}\\frac{Q_2(n(n+1)\\cdots(n+\\ell))}{n^2}\\]infinite? If $\\ell\\geq 2$ then is\\[\\lim_{n\\to \\infty}\\frac{Q_2(n(n+1)\\cdots(n+\\ell))}{n^{\\ell+1}}=0?\\]","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A057521","name":"Powerful part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n = b*c^2*d^3 then a(n) = c^2*d^3 when b is minimized.","terms":"1,1,1,4,1,1,1,8,9,1,1,4,1,1,1,16,1,9,1,4,1,1,1,8,25,1,27,4,1,1,1,32,1,1,1,36,1,1,1,8,1,1,1,4,9,1,1,16,49,25,1,4,1,27,1,8","url":"https://oeis.org/A057521"},{"id":"A389244","name":"Powerful part of n*(n+1): a(n) = A057521(n*(n+1)).","terms":"1,1,4,4,1,1,8,72,9,1,4,4,1,1,16,16,9,9,4,4,1,1,8,200,25,27,108,4,1,1,32,32,1,1,36,36,1,1,8,8,1,1,4,36,9,1,16,784,1225,25","url":"https://oeis.org/A389244"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}