PlanetMath: deficiency

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deficiency

(Definition)

Given an integer with divisors $d_1, \ldots , d_k$ (where the divisors are in ascending order and , ) the difference

$\displaystyle 2n - \left( \sum_{i = 1}^k d_i \right)$

is the deficiency of

. Or if one prefers,

$\displaystyle n - \left( \sum_{i = 1}^{k - 1} d_i \right).$

The deficiency is essentially the same thing as the abundance multiplied by

. Thus, the deficiency is positive for deficient numbers, 0 for perfect numbers and negative for abundant numbers.

For example, the divisors of 13 add up to 14, which is 12 less than 26. Therefore, 12 has an deficiency of 12. Another example: the divisors of 14 add up to 24, which is 4 less than 28. The deficiency of the first 72 integers is listed in A033879 of Sloane's OEIS.

"deficiency" is owned by PrimeFan.

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See Also: abundance

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Cross-references: OEIS, abundant numbers, negative, perfect numbers, deficient numbers, positive, abundance, difference, ascending order, divisors, integer
There are 3 references to this entry.

This is version 3 of deficiency, born on 2007-03-03, modified 2007-03-03.
Object id is 9011, canonical name is Deficiency.
Accessed 676 times total.

Classification:

AMS MSC:

11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors)

Pending Errata and Addenda

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