Problem 18

Erdős Problem #18: declared status 'open'. Formalized: yes. We call $m$ practical if every integer $n<m$ 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$$h(m) < (\log\log m)^{O(1)}?$$Is it true that $h(n!)<n^{o(1)}$? Or perhaps even $h(n!)<(\log n)^{O(1)}$? Prize: no. OEIS: A005153. Tags: divisors, factorials, number theory.