# abundance

Given an integer $n$ with divisors $d_{1},\ldots,d_{k}$ (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 Abundance 2013-03-22 16:05:49 2013-03-22 16:05:49 CompositeFan (12809) CompositeFan (12809) 9 CompositeFan (12809) Definition msc 11A05 Deficiency