Do Quasiperfect Numbers Exist?
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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
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): $16$