abundance

Given an integer $n$ with divisors $d_1,\mathrm{\dots },d_k$ (where the divisors are in ascending order and $d_1=1$, $d_k=n$) the difference

$$\left(\sum _{i=1}^k{d}_i\right)-2n$$

is the abundance of $n$. Or if one prefers,

$$\left(\sum _{i=1}^{k-1}{d}_i\right)-n.$$

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