Primitive Abundant Number/Examples/748

Example of Primitive Abundant Number

$748$ is a primitive abundant number:

$1 + 2 + 4 + 11 + 17 + 22 + 34 + 44 + 68 + 187 + 374 = 764 > 748$

Proof

From $\sigma_1$ of $748$, we have:

$\map {\sigma_1} {748} - 748 = 764$

where $\sigma_1$ denotes the divisor sum function.

Thus, by definition, $748$ is abundant number.

The aliquot parts of $748$ are enumerated at $\sigma_0$ of $748$:

$1, 2, 4, 11, 17, 22, 34, 44, 68, 187, 374$

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

Hence the result, by definition of primitive abundant number.

$\blacksquare$