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:
There are infinitely many deficient numbers.
For example, pk, 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.
|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|
Deficient and abundant numbers were first so named
in Nicomachus' Introductio Arithmetica (c. 100 ad).
See Also: AmicableNumber, SigmaFunction