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:
 * $\omega \left({n}\right) \ge 7$

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

Also see

 * Quasiperfect Number is Square of Odd Integer