Problem Statement
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 $h(n!)<(\log n)^{O(1)}$?
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 $h(n!)<(\log n)^{O(1)}$?
Categories:
Number Theory Divisors Factorials
Progress
It is easy to see that almost all numbers are not practical. Erdős originally showed that $h(n!) <n$. Vose [Vo85] proved the existence of infinitely many practical $m$ such that $h(m)\ll (\log m)^{1/2}$.The sequence of practical numbers is A005153 in the OEIS.
The reward of \$250 is offered in [Er81h], apparently (although this is not entirely clear) for a proof or disproof of whether\[h(n!) <(\log n)^{O(1)}.\]See also [304] and [825].
Source: erdosproblems.com/18 | Last verified: January 13, 2026