Is the aliquot sum of an odd number always odd?

Question

I am thinking yes. Here's my reasoning:

Let $s(x)$ be the aliquot sum of $x$.

If $p$ is prime then $s(px) = s(x) + ps(x) + x$

Base Case: if $x$ is an odd prime, then $s(x) = 1$

Assume it is true up to $x$ so that $s(x)$ is odd.

Inductive Case: $s(px) = s(x) + ps(x) + x =$ odd + odd + odd = odd.

I checked the Wikipedia article on aliquot sum, but I didn't see this as a property.

Answer 1

Square numbers are a special case.