Primitive Semiperfect Number/Examples/1190

Example of Primitive Semiperfect Number

$1190$ is a primitive semiperfect number:

$1 + 2 + 5 + 10 + 14 + 17 + 34 + 70 + 85 + 119 + 238 + 595 = 1190$

Proof

First it is demonstrated that $1190$ is semiperfect.

The aliquot parts of $1190$ are enumerated at $\sigma_0$ of $1190$:

$1, 2, 5, 7, 10, 14, 17, 34, 35, 70, 85, 119, 170, 238, 595$

$1190$ is the sum of a subset of its aliquot parts:

$1 + 2 + 5 + 10 + 14 + 17 + 34 + 70 + 85 + 119 + 238 + 595$

Thus $1190$ 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.

$\blacksquare$