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[parent] abundance (Definition)

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

$\displaystyle \left( \sum_{i = 1}^k d_i \right) - 2n$
is the abundance of $ n$. Or if one prefers,
$\displaystyle \left( \sum_{i = 1}^{k - 1} d_i \right) - n.$

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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Cross-references: deficient numbers, negative, almost perfect number, perfect number, quasiperfect number, abundant numbers, numbers, positive, OEIS, difference, ascending order, divisors, integer
There are 7 references to this entry.

This is version 6 of abundance, born on 2006-07-20, modified 2007-03-06.
Object id is 8159, canonical name is Abundance.
Accessed 1187 times total.

Classification:
AMS MSC11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors)

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