, a deficient number
or defective number
is a number n
for which σ
) < 2n
. Here σ
) is the sum-of-divisors function
: the sum of all positive divisors
, 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
) is called the deficiency
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, …
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).