Problem Statement
Call a number $k$-perfect if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\log\log n)$?
Categories:
Number Theory
Progress
A question of Erdős, as reported in problem B2 of Guy's collection [Gu04]. Guy further writes 'It has even been suggested that there may be only finitely many $k$-perfect numbers with $k\geq 3$.' The largest $k$ for which a $k$-perfect number has been found is $k=11$ - see this page for more information.These are known as multiply perfect numbers. When $k=2$ this is the definition of a perfect number.
Source: erdosproblems.com/1053 | Last verified: January 19, 2026