deficient number

Suppose you take a positive integer n and add its positive divisors.  For example, if n=18, then the sum is 1 + 2 + 3 + 6 + 9 + 18 = 39.  In general, when we do this with n one of the following three things happens:

the sum is	and we say n is a	examples
less than 2n	deficient number	1, 2, 3, 4, 5, 8, 9
equal to 2n	perfect number	6, 28, 496
greater than 2n	abundant number	12, 18, 20, 24, 30

There are infinitely many deficient numbers.  For example, p^k, with p any prime and k > 0, is deficient.  Also if n is any perfect number, and d divides n (where 1 < d < n), then d is deficient.

Deficient and abundant numbers were first so named in Nicomachus' Introductio Arithmetica (c. 100 ad).

See Also: AmicableNumber, SigmaFunction