I was wondering, if there is a formula to determine how many odd composite pairs there are until a given $n\in\mathbb{Z}^+$ like $(25,27), (33,35), etc.$ Theoretically it can be calculated, because they consist of prime factors, and we do know how many primes there are $\pi(n)$.
The number of twin composite pairs should somehow be calculated with a combination formula, I am just wondering of there are some research on that.
It is clear that there are infinitely many such odd composite pairs (for eg. take $(15k+3,15k+5)$ for any $k \in \mathbb{N}$). To find an asymptotic formula, let $\pi(x)$ be the prime counting function. Either a pair contains atleast one prime or a pair contains both composites. There are roughly $\frac{x}{2}$ pairs less than $x$, of which $\pi_2(x)$ are twin prime pairs. Mixed pairs have a count of $2\pi(x)-2\pi_2(x)$ (Two $\pi_2(x)$ are subtracted since twin pairs are to be removed, which are counted twice when we choose $(p-2,p)$, $(p,p+2)$ for every prime $p$). Thus, if $f(x)$ is the number of odd composite pairs less than $x$- $$f(x) \approx\frac{x}{2}-\pi_2(x)-(2\pi(x)-2\pi_2(x)) \approx \frac{x}{2}-2\pi(x)+\pi_2(x)$$ Thus: $$f(x) \sim \frac{x}{2}-\frac{2x}{\log x}+\frac{x}{\log^2 x} \implies f(x) \sim \frac{x}{2}$$