Primitive Abundant Number/Examples/650

Example of Primitive Abundant Number

$650$ is a primitive abundant number:

$1 + 2 + 5 + 10 + 13 + 25 + 26 + 50 + 65 + 130 + 325 = 652 > 650$

Proof

From $\sigma_1$ of $650$, we have:

$\map {\sigma_1} {650} - 650 = 652$

where $\sigma_1$ denotes the divisor sum function: the sum of all divisors of $650$.

Thus, by definition, $650$ is an abundant number.

The aliquot parts of $650$ are enumerated at $\sigma_0$ of $650$:

$1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325$

By inspecting the divisor sum of each of these, they are seen to be deficient.

Hence the result, by definition of primitive abundant number.

$\blacksquare$