Given an integer n with divisorsMathworldPlanetmathPlanetmath d1,,dk (where the divisors are in ascending order and d1=1, dk=n) the difference


is the abundance of n. Or if one prefers,


For example, the divisors of 12 (which are 1, 2, 3, 4, 6 and 12) add up to 28, which is 4 more than 24 (twice 12). Therefore, 12 has an abundance of 4. For the sake of comparison, the divisors of 13 are 1 and 13, adding up to 14, which is 12 less than 26 (twice 13). Therefore, 13 has an abundance of -12. A033880 in Sloane’s OEIS lists the abundance of the first sixty-three positive integers.

Thus numbers with positive abundance are abundant numbers. A number with an abundance of exactly 1 is a quasiperfect number, while a number with 0 abundance is a perfect number. A number with an abundance of -1 is an almost perfect number (this is true of all powers of 2); all numbers with negative abundance are deficient numbers.

Title abundance
Canonical name Abundance
Date of creation 2013-03-22 16:05:49
Last modified on 2013-03-22 16:05:49
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 9
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A05
Related topic Deficiency