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 isand we say n is aexample
less than 2ndeficient number1, 2, 3, 4, 5, 7, 8, 9
equal to 2nperfect number6, 28, 496
greater than 2nabundant number12, 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


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