# abundant number

Suppose you take a positive integer *n* and add its positive
proper divisors (denoted σ(*n*). For example, if *n* is 12, then the sum is
σ(*n*) = 1 + 2 + 3 + 4 + 6 = 16. When we do this with the integer
*n* one of the following three things happen:

the sum is | and we say n is a | example |
---|---|---|

less than 2n | deficient number | 1, 2, 3, 4, 5, 7, 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).

There are infinitely many abundant numbers, both
even (e.g., every multiple of 12) and
odd (e.g., every odd multiple of 945). Every
proper multiple of a perfect number, and every
multiple of an abundant
number, is
abundant (because when *n* > 1,
σ(*n*)/*n* > 1+1/*n*; and σ
is a multiplicative function). Deleglise
has shown that on the average 24.7% of
the positive integers are abundant (more
specifically, that the natural density of the
abundant integers is in the open interval
(0.2474, 0.2480)).

Every integer greater than 20161 can be written as the sum of two abundant numbers.

**See Also:** AmicableNumber, AliquotSequence

**References:**

- Deleglise98
M. Deléglise, "Bounds for the density of abundant integers,"Experimental Math.,7:2 (1998) 137--143.MR 2000a:11137(Abstract available)