All Random Solved Random Open
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 $h(n!)<n^{o(1)}$?

It is easy to see that almost all numbers are not practical. This may be true with $m=n!$. Erdős originally showed that $h(n!) <n$. Vose [Vo85] has shown that $h(n!)\ll n^{1/2}$.

Related to [304].