# 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 nis aexamples less than 2 ndeficient number1, 2, 3, 4, 5, 8, 9 equal to 2 nperfect number 6, 28, 496 greater than 2 nabundant 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

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