Do Quasiperfect Numbers Exist?

Open Question
It is not known whether any quasiperfect numbers actually exist.

Progress
As of January $2014$, there are no known quasiperfect numbers.

It was proved by and  that if $n$ is a quasiperfect number, then:
 * $\map \omega n \ge 7$

and:
 * $n > 10^{35}$

where $\map \omega n$ denotes the number of distinct prime factors of $n$.

Also see

 * Quasiperfect Number is Square of Odd Integer