Primitive Semiperfect Number/Examples/350

Example of Primitive Semiperfect Number

$350$ is a primitive semiperfect number:

$1 + 5 + 10 + 14 + 25 + 50 + 70 + 175 = 350$

Proof

First it is demonstrated that $350$ is semiperfect.

The aliquot parts of $350$ are enumerated at $\sigma_0$ of $350$:

$1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175$

$350$ is the sum of a subset of its aliquot parts:

$1 + 5 + 10 + 14 + 25 + 50 + 70 + 175 = 350$

Thus $350$ 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.

$\blacksquare$