Divisibility-based sets of integers

Forms of factorization:

Prime number

Composite number

Powerful number

Square-free number

Achilles 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

Numbers with many divisors:

Abundant number

Highly abundant number

Superabundant number

Colossally abundant number

Highly composite number

Superior highly composite number

Other:

Untouchable number

Deficient number

Weird number

Amicable number

Friendly number

Sociable number

Solitary number

Sublime number

Harmonic divisor number

Frugal number

Equidigital number

Extravagant number

See also:

Divisor function

Divisor

Prime factor

Factorization

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In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. Here σ(n) is the sum-of-divisors function: the sum of all positive divisors of n, including n itself. An equivalent definition is that the sum of all proper divisors of the number (divisors other than the number itself) is less than the number. The value 2n − σ(n) is called the deficiency of n.

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.

An infinite number of both even and odd deficient numbers exist. For example, all odd numbers with one or two distinct prime factors, and all proper divisors of deficient or perfect numbers are deficient.

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).

[edit] See also

• Almost perfect number
• Amicable number
• Sociable number

[edit] External links

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