Primitive Semiperfect Number/Examples/490

Example of Primitive Semiperfect Number

$490$ is a primitive semiperfect number:

$2 + 5 + 7 + 14 + 49 + 70 + 98 + 245 = 490$

Proof

First it is demonstrated that $490$ is semiperfect.

The aliquot parts of $490$ are enumerated at $\sigma_0$ of $490$:

$1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245$

$490$ is the sum of a subset of its aliquot parts:

$2 + 5 + 7 + 14 + 49 + 70 + 98 + 245 = 490$

Thus $490$ is semiperfect by definition.

By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient except for $70$.

By Semiperfect Number is not Deficient, none of the deficient aliquot parts are themselves semiperfect.

As for $70$ itself, it is seen to be a weird number.

So, by definition, $70$ is not semiperfect.

Hence the result, by definition of primitive semiperfect number.

$\blacksquare$