Primitive Semiperfect Number/Examples/910

Example of Primitive Semiperfect Number

$910$ is a primitive semiperfect number:

$1 + 2 + 5 + 7 + 10 + 13 + 26 + 35 + 65 + 70 + 91 + 130 + 455 = 910$

Proof

First it is demonstrated that $910$ is semiperfect.

The aliquot parts of $910$ are enumerated at $\sigma_0$ of $910$:

$1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455$

$910$ is the sum of a subset of its aliquot parts:

$1 + 2 + 5 + 7 + 10 + 13 + 26 + 35 + 65 + 70 + 91 + 130 + 455$

Thus $910$ 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.

$\blacksquare$