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In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number n such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n) = σ(n) − n, then being equal to n − 1. The only known almost perfect numbers are powers of 2 with non-negative exponents (sequence A000079 in the OEIS). Therefore the only known odd almost perfect number is 2 = 1, and the only known even almost perfect numbers are those of the form 2 for some positive integer k; however, it has not been shown that all almost perfect numbers are of this form. It is known that an odd almost perfect number greater than 1 would have at least six prime factors.

If m is an odd almost perfect number then m(2m − 1) is a Descartes number. Moreover if a and b are positive odd integers such that ${\displaystyle b+3<a<{\sqrt {m/2}}}$ and such that 4m − a and 4m + b are both primes, then m(4m − a)(4m + b) would be an odd weird number.

See also

• Perfect number
• Quasiperfect number

References

1. Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which 2−10 < σ(N)/N < 2+10" (PDF). Mathematics of Computation. 32: 303–309. doi:10.2307/2006281. ISSN 0025-5718. JSTOR 2006281. MR 0485658. Zbl 0376.10005.
2. Kishore, Masao (1981). "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation. 36 (154): 583–586. doi:10.2307/2007662. ISSN 0025-5718. JSTOR 2007662. Zbl 0472.10007.
3. Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip (2008). "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (eds.). Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006. CRM Proceedings and Lecture Notes. Vol. 46. Providence, RI: American Mathematical Society. pp. 167–173. ISBN 978-0-8218-4406-9. Zbl 1186.11004.
4. Melfi, Giuseppe (2015). "On the conditional infiniteness of primitive weird numbers". Journal of Number Theory. 147: 508–514. doi:10.1016/j.jnt.2014.07.024.

Further reading

• Guy, R. K. (1994). "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers". Unsolved Problems in Number Theory (2nd ed.). New York: Springer-Verlag. pp. 16, 45–53.
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. p. 110. ISBN 1-4020-4215-9. Zbl 1151.11300.
• Sándor, Jozsef; Crstici, Borislav, eds. (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 37–38. ISBN 1-4020-2546-7. Zbl 1079.11001.
• Singh, S. (1997). Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker. p. 13. ISBN 9780802713315.

External links

• Weisstein, Eric W. "Almost perfect number". MathWorld.

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

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