Primitive Abundant Number/Examples/272

Example of Primitive Abundant Number

$272$ is a primitive abundant number:

$1 + 2 + 4 + 8 + 16 + 17 + 34 + 68 + 136 = 286 > 272$

Proof

From $\sigma_1$ of $272$, we have:

$\map {\sigma_1} {272} - 272 = 286$

where $\sigma_1$ denotes the divisor sum function: the sum of all divisors of $272$.

Thus, by definition, $272$ is an abundant number.

The aliquot parts of $272$ are enumerated at $\sigma_0$ of $272$:

$1, 2, 4, 8, 16, 17, 34, 68, 136$

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

Hence the result, by definition of primitive abundant number.

$\blacksquare$