# A question regarding the amount of composite solutions for a equation

Consider that for some natural value of $$x$$, we ask, how many values of $$k$$ are there, such that $$x-k$$ and $$k$$ are both composite numbers?

Now, consider that $$x=\prod_{i=1}^r p_i$$ where $$p_i$$ is the $$i$$th prime factor of $$x$$. And now, consider that by saying $$k=cp_l$$ where $$l\leq r$$, we have that $$x-k$$ is composite and that $$k$$ is composite. But still, this is only a bit of the values of it. So, if we say that $$Q(x)$$ is the amount of solutions for this, then, we can say that: $$Q(x)\approx\sum_{p|x} (\lfloor {x \over p} \rfloor - 1) - \sum_{p_1,p_2|x\;p_1>p_2} \lfloor {x \over p_1p_2} \rfloor + \cdots$$

So, can someone help me find a better way of writing $$Q(x)$$?

• so it is somewhat goldbach related ... – user645636 Jan 22 '20 at 20:55

$$Q(n) = A073610(n) + n - 3 - 2 \pi(n-2)$$ for $$n\ge 3$$, where OEIS sequence A073610(n) gives the number of $$k$$ such that $$k$$ and $$n-k$$ are prime.