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In number theory, a deficient or deficient number is a number n for which the sum of divisors σ(n)<2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)<n. The value 2n − σ(n) (or n − s(n)) is called the number's deficiency.

Examples

The first few deficient numbers are:

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, … (sequence A005100 in OEIS)

As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, and their sum is 32. Because 32 is less than 2 × 21, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.
Properties

An infinite number of both even and odd deficient numbers exist
All odd numbers with one or two distinct prime factors are deficient
All proper divisors of deficient or perfect numbers are deficient.
There exists at least one deficient number in the interval $$[n, n + (\log n)^2]$$ for all sufficiently large n.[1]

Related concepts

Closely related to deficient numbers are perfect numbers with σ(n) = 2n, and abundant numbers with σ(n) > 2n. The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100).

Almost perfect number
Amicable number
Sociable number

References

Sándor et al (2006) p.108

Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.

The Prime Glossary: Deficient number
Weisstein, Eric W., "Deficient Number", MathWorld.
deficient number at PlanetMath.org.

It is unknown whether there exists a prime number p such that Cp is also prime.

As of February 2012, the largest known generalized Cullen prime is 427194 × 113 427194 + 1. It has 877,069 digits and was discovered by a PrimeGrid participant from United States.[3]

150 = 2 × 3 × 52

Mathematics Encyclopedia