Deficiency - Definition and Overview

In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. Here σ(n) is the divisor function: the sum of all positive divisors of n, including n itself. The value 2n − σ(n) is 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, ... (sequence A005100 in OEIS).

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

See also

• abundant number
• amicable number
• sociable number

External links

• The Prime Glossary: Deficient number (http://primes.utm.edu/glossary/page.php?sort=DeficientNumber)
• MathWorld: Deficient number (http://mathworld.wolfram.com/DeficientNumber.html)