Primitive Semiperfect Number/Examples/770

Example of Primitive Semiperfect Number

$770$ is a primitive semiperfect number:

$1 + 5 + 7 + 11 + 14 + 35 + 55 + 70 + 77 + 110 + 385 = 770$

Proof

First it is demonstrated that $770$ is semiperfect.

The aliquot parts of $770$ are enumerated at $\sigma_0$ of $770$:

$1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385$

$770$ is the sum of a subset of its aliquot parts:

$1 + 5 + 7 + 11 + 14 + 35 + 55 + 70 + 77 + 110 + 385$

Thus $770$ 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.

$\blacksquare$