Difficult looking summation problem

Question

$$\sum_{n=1}^{\infty} \omega(n)(x^n - 2x^{2n} + (-x)^n) = \frac{2x^2}{1-x^4} $$ Where $\omega(n)$ is the number of prime factors of $n$ and $\vert x \vert < 1$

Answer 1

Hint: Writing $f(n)=\omega(n)(x^n-2x^{2n}+(-x)^n)$, rearrange as $$ \sum_{n=1}^\infty f(n) = \sum_{k\text{ odd}}\; \sum_{m=0}^\infty f(2^mk) $$ The inner sum then telescopes, taking the $\omega(k)$ with it and leaving just $$ \sum_{k\text{ odd}} 2x^{2k} $$