{"schema":"vela.problem-packet.v0.1","problem":391,"statement":"Let $t(n)$ be maximal such that there is a representation\\[n!=a_1\\cdots a_n\\]with $t(n)=a_1\\leq \\cdots \\leq a_n$. Obtain good bounds for $t(n)/n$. In particular, is it true that\\[\\lim \\frac{t(n)}{n}=\\frac{1}{e}?\\]Furthermore, does there exist some constant $c&#62;0$ such that\\[\\frac{t(n)}{n} \\leq \\frac{1}{e}-\\frac{c}{\\log n}\\]for infinitely many $n$?","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A034258","name":"Write n! as a product of n numbers, n! = k(1)*k(2)*...*k(n) with k(1) <= k(2) <= ..., in all possible ways; a(n) = max value of k(1).","terms":"1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,8,8,8,9,9,10,10,10,10,11,11,12,12,12,12,12,12,13,13,13,14,14,15,","url":"https://oeis.org/A034258"},{"id":"A034259","name":"Smallest m such that A034258(m) >= n.","terms":"1,4,9,14,16,20,24,27,32,34,38,40,46,49,51,57,58,62,65,68,72,77,80,84,87,90,93,94,100,104,108,111,114,115,118,125,125,128","url":"https://oeis.org/A034259"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}