A problem of Benkoski and Erdős. In other words, this problem asks for an upper bound for the abundancy index of weird numbers. This could be true with $C=3$. We must have $C>2$ since $\sigma(70)=144$ but $70$ is not the distinct sum of integers from $\{1,2,5,7,10,14,35\}$.

Erdős suggested that as $C\to \infty$ only divisors at most $\epsilon n$ need to be used, where $\epsilon \to 0$.

See also [18].