Primitive Semiperfect Number/Examples/945

Example of Primitive Semiperfect Number
$945$ is a primitive semiperfect number:
 * $1 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 945$

Proof
First it is demonstrated that $945$ is semiperfect.

The aliquot parts of $945$ are enumerated at $\tau$ of $945$:
 * $1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315$

$945$ is the sum of a subset of its aliquot parts:
 * $1 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 63 + 105 + 135 + 189 + 315$

Thus $945$ is semiperfect by definition.

By inspecting the $\sigma$ values of each of those aliquot parts, they are seen to be deficient.

By Semiperfect Number is not Deficient, none of these are themselves semiperfect.

Hence the result, by definition of primitive semiperfect number.