Deficient number

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 (sequence A005100 in OEIS) are:

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, …

As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, whose 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 prime numbers, all prime powers 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).

Properties

Every integer of the form 12m+1 up to 10⁷ is deficient. However, the product of consecutive primes P = 5 × 7 × 11 × 13 × 17 × … × 107 × 109, which lies between 10⁴³ and 10⁴⁴, is of the form 12m+1 and is abundant (therefore not deficient). This can be seen from the fact that it has divisors P, P/5, P/7, ... P/109 and 1+1/5+1/7+...+1/109 > 2.

See also

• Abundant number
• Almost perfect number
• Amicable number
• Sociable number

External links

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