Almost perfect number

In math, an almost perfect number (also called slightly defective or least deficient number) is a type of natural number n. The sum of n's divisors must be equal to 2n − 1. Every known almost perfect number is a power of 2 and has non-negative exponents (sequence A000079 in the OEIS).

Examples

For example, the divisors of 32 are 1, 2, 4, 8, 16 and 32. The sum of those is 63. 32 ⋅ 2 - 1 is 63. This makes 32 an almost perfect number.

Odd numbers

The only known odd almost perfect number 1. An odd almost perfect number that's not 1 is possible. It would, however, have to have six prime factors.^[1]^[2]

References

↑ Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which 2−10⁻¹² < σ(N)/N < 2 10⁻¹²" (PDF). Mathematics of Computation. 32: 303–309. doi:10.2307/2006281. ISSN 0025-5718. JSTOR 2006281. MR 0485658. Zbl 0376.10005.
↑ 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.

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.

[Kis1978-1] Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which 2−10⁻¹² < σ(N)/N < 2 10⁻¹²" (PDF). Mathematics of Computation. 32: 303–309. doi:10.2307/2006281. ISSN 0025-5718. JSTOR 2006281. MR 0485658. Zbl 0376.10005.

[Kis1981-2] 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.

[1]

[2]

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 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 Descartes Refactorable Superperfect

Examples

Odd numbers

References

Further reading