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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics the divisor function σa(n) is defined as the sum of the ath powers of the divisors of n, or The notations d(n) and (the tau function) are also used to denote σ0(n), or the number of divisors of n. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

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). Nicomachus (c. ... -1... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

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. In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, excluding itself. ...

See also

• abundant number
• amicable number
• sociable number

In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ... Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ... Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ...

External links

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