Quasiperfect number

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 10³⁵ and have at least seven distinct prime factors. ^[1]

Notes

1. ^ Hagis, Peter; Cohen, Graeme L. (1982). "Some results concerning quasiperfect numbers". J. Austral. Math. Soc. Ser. A 33 (2): 275–286. doi:10.1017/S1446788700018401. MR 0668448. 

References

• Brown, E.; Abbott, H.; Aull, C.; Suryanarayana, D. (1973). "Quasiperfect numbers". Acta Arithm. 22: 439–447. MR 0316368. http://matwbn.icm.edu.pl/ksiazki/aa/aa22/aa2245.pdf. 
• Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which...". Math. Comput. 32: 303–309. MR 0485658. 
• Cohen, Graeme L. (1980). "On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers". J. Austral. Math. Soc. 29: 369–384. doi:10.1017/S1446788700021376. MR 0569525. 
• James J. Tattersall (1999). Elementary number theory in nine chapters. Cambridge University Press. pp. 147. ISBN 0-521-58531-7. 

v t e Divisibility-based sets of integers Overview Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Forms of factorization Prime number Composite number Semiprime number Pronic number Sphenic number Square-free number Powerful number Perfect power Achilles number Smooth number Regular number Rough number Unusual 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 With many divisors Abundant number Primitive abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Weird number Related to aliquot sequences Untouchable number Amicable number Sociable number Other sets Deficient number Friendly number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number

This number theory-related article is a stub. You can help Wikipedia by expanding it. v t e