{"schema":"vela.problem-packet.v0.1","problem":727,"statement":"Let $k\\geq 2$. Does\\[(n+k)!^2 \\mid (2n)!\\]for infinitely many $n$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A002503","name":"Numbers k such that binomial(2*k,k) is divisible by (k+1)^2.","terms":"5,14,27,41,44,65,76,90,109,125,139,152,155,169,186,189,203,208,209,219,227,230,237,265,275,298,307,311,314,321,324,329,3","url":"https://oeis.org/A002503"},{"id":"A343507","name":"a(n) is the smallest nonnegative integer k such that (2*k)! / (k+n)!^2 is an integer.","terms":"0,208,3475,8174,252965,3648835,72286092,159329607,2935782889","url":"https://oeis.org/A343507"},{"id":"A389396","name":"Numbers k such that binomial(2*k,k) is divisible by (k+1)^2*(k+2)^2.","terms":"208,458,987,1220,1455,1597,1889,2012,2144,2330,2477,2663,2991,3353,3415,3430,3439,3475,3476,3551,3563,3568,3617,3625,372","url":"https://oeis.org/A389396"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}