Problem 825

Erdős Problem #825: declared status 'proved'. Formalized: yes. Is there an absolute constant $C>0$ such that every integer $n$ with $\sigma(n)>Cn$ is the distinct sum of proper divisors of $n$? Current best: In other words, this problem asks for an upper bound for the abundancy index of weird numbers. A similar argument shows that either there are infinitely many primitive weird numbers or there is an upper bound for the abundancy index of all weird numbers. Prize: $25. OEIS: A006037, A330244. Tags: number theory.