{"schema":"vela.problem-packet.v0.1","problem":367,"statement":"Let $B_2(n)$ be the $2$-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for every fixed $k\\geq 1$,\\[\\prod_{n\\leq m<n+k}B_2(m) \\ll n^{2+o(1)}?\\]Or perhaps even $\\ll_k n^2$?","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"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}