How to express this sum of certain non-divisors in a different form?

Question

I need help finding a way to express this sum. $$\sum_{\substack{b\space\nmid\space x\\{\left\lfloor{\frac{x}{b}}\right\rfloor}\space\text{is even}}}^xb$$ I need to express this in a different form that isn't based on such conditions. I know that $\sum_{b\space\nmid\space x}^x=\frac{x^2+x}{2}-\sigma(x)$, but how could you extend to the condition $\left\lfloor\frac{x}{b}\right\rfloor\text{is even}$?