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and the aliquot sum s(12) of proper divisors is:

:

?_-1(n) is sometimes called the abundancy index of n, and we have:

:

Table of values

The cases x = 2 to 5 are listed in through , x = 6 to 24 are listed in through .

Properties

Formulas at prime powers

For a prime number p,

:

because by definition, the factors of a prime number are 1 and itself. Also, where

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