In number theory, a deficient number or defective number is a number n for which the sum of divisors σ(n)', 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[edit]

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[edit]

• 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 for all sufficiently large n.^[1]

Related concepts[edit]

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).

See also[edit]

• Almost perfect number
• Amicable number
• Sociable number

References[edit]

• 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. 

External links[edit]

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

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