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 Peter Hagis, Jr. and Graeme L. Cohen 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

Sources

• 1982: Peter Hagis, Jr. and Graeme L. Cohen: Some results concerning quasiperfect numbers (J. Austral. Math. Soc. Ser. A Vol. 33, no. 2: pp. 275 – 286)

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $16$
• 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $16$