Erdős Problems

Tags Prizes

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 $h(n!)<n^{o(1)}$? Or perhaps even $h(n!)<(\log n)^{O(1)}$?

#18: [Er74b][Er79][ErGr80][Er95][Er96b][Er98]

number theory, divisors, factorials

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.