Quasiperfect number

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Divisibility-based sets of integers

Form of factorization:

Prime number

Composite number

Powerful number

Square-free number

Achilles number

Constrained divisor sums:

Perfect number

Almost perfect number

Quasiperfect number

Multiply perfect number

Hyperperfect number

Superperfect number

Unitary perfect number

Semiperfect number

Primitive semiperfect number

Practical number

Numbers with many divisors:

Abundant number

Highly abundant number

Superabundant number

Colossally abundant number

Highly composite number

Superior highly composite number

Other:

Deficient number

Weird number

Amicable number

Friendly number

Sociable number

Solitary number

Sublime number

Harmonic divisor number

Frugal number

Equidigital number

Extravagant number

See also:

Divisor function

Divisor

Prime factor

Factorization

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In mathematics, a quasiperfect number is a theoretical natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Quasiperfect numbers are abundant numbers.

No quasiperfect numbers have been found so far, but if a quasiperfect number exists, it must be an odd square number greater than 10³⁸ and have at least seven distinct prime factors. ^[1]

[edit] Notes

1. ^ Hagis, Peter & Cohen, Graeme L., (1982). "Some results concerning quasiperfect numbers". J. Austral. Math. Soc. Ser. A 33: 275–286. doi:10.1017/S1446788700018401. 

[edit] References

• James J. Tattersall (1999). Elementary number theory in nine chapters. Cambridge University Press. pp. 147. ISBN 0521585317.