Problem 1053

Erdős Problem #1053: declared status 'open'. Formalized: no. Call a number $k$-perfect if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\log\log n)$? Current best: 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. Prize: no. OEIS: A007539. Tags: number theory.