OPEN
This is open, and cannot be resolved with a finite computation.
- $250
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 $h(n!)<(\log n)^{O(1)}$?
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].
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This page was last edited 31 October 2025.
Additional thanks to: Dogmachine and Ralf Stephan
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #18, https://www.erdosproblems.com/18, accessed 2026-01-16
