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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 $Q(x)$?

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    $\begingroup$ so it is somewhat goldbach related ... $\endgroup$ – user645636 Jan 22 '20 at 20:55
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$Q(n) = A073610(n) + n - 3 - 2 \pi(n-2)$ for $n\ge 3$, where OEIS sequence A073610(n) gives the number of $k$ such that $k$ and $n-k$ are prime.

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  • $\begingroup$ Now added to OEIS as sequence A331622. $\endgroup$ – Robert Israel Jan 23 '20 at 13:20

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