11A05-Abundance.tex

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\begin{document}
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 {\em 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.
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