Deficient number

In mathematics, a deficient number (also sometimes called a defective number) is a number n for which the sum of all its positive divisors (including n, the divisor function, σ(n)) is less than 2n; the value 2n - σ(n) is sometimes called the deficiency of n.

Deficient numbers were first introduced in Nicomachus' Introductio Arithmetica (circa 100). The first few deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (Sloane's A005100).

An infinite number of both even and odd deficient numbers exist; for example, all prime numbers, prime powers and all proper divisors of deficient or perfect numbers are deficient.

Also see

• Perfect number
• Abundant number
• Amicable number
• Sociable number

External links

• The Prime Glossary: Deficient number
• MathWorld: Deficient number