{"schema":"vela.problem-packet.v0.1","problem":18,"statement":"We call $m$ practical if every integer $1\\leq n&#60;m$ is the sum of distinct divisors of $m$. If $m$ is practical then let $h(m)$ be such that $h(m)$ many divisors always suffice.Are there infinitely many practical $m$ such that\\[h(m) &#60; (\\log\\log m)^{O(1)}?\\]Is it true that $h(n!)&#60;n^{o(1)}$? Or perhaps even $h(n!)&#60;(\\log n)^{O(1)}$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A005153","name":"Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers.","terms":"1,2,4,6,8,12,16,18,20,24,28,30,32,36,40,42,48,54,56,60,64,66,72,78,80,84,88,90,96,100,104,108,112,120,126,128,132,140,14","url":"https://oeis.org/A005153"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}