Q:

Function for counting composite numbers

I am analyzing the Goldbach conjecture for prime numbers, which states that every even number is a sum of two prime numbers.
I have a question about counting of composite numbers. Is there any known function to count couples of composite numbers?
I am interesting in counting of how many composite couples fulfill:
$$\text{composite}_1+\text{composite}_2=2^n$$
For example, $64 = 3*3+5*11 = 3*5+7*7 = 5*5+3*13$ ,there are three couples and $f(64)=3$.
For $32$ there are no composite sums couples so $f(32)=0$.
For $128 = 3*3*7+5*13 = 3*17+7*11 = 5*7+3*31 = 3*11+5*19 = 3*3+7*17$ there are $5$ couples so $f(128)=5$
Function itself may be approximate, similar as the prime number counting function $x/ln(x)$.
The function should not count composite couples of $2$, because this can be counted by $(2^n/4)-1$.
Such function may lead to a proof of the Goldbach conjecture for prime sums equals to $2^n$ and maybe even a proof of the whole Goldbach conjecture. 

A:

Let $x = 2^n$ for convenience.
There are $x/4$ ways to write $x$ as a sum of two odd numbers.  Of these, $\pi(x/2)$ have a prime (or $1$) in the first (lesser) component, and $\pi(x) - \pi(x/2)$ have a prime in the second (greater) component.  Therefore the number of pairs of odd composite numbers summing to $x$ is bounded below by
$$x/4 - \pi(x) \sim x/4$$
and bounded above by
$$x/4 - \pi(x/2) \sim x/4,$$
so it is clearly asymptotic to $x/4$ with an error term of size $\Theta(x/\log x)$.  The relative error in this approximation is $\Theta(1/\log x)$, exactly the same order of accuracy as $\pi(x) \approx x/\log x$ which has an error of size $\Theta(x/\log^2 x)$.
Addendum: for an even better approximation, one can prove via sieve methods that the number of prime pairs summing to $x$ is bounded above by $O(x/\log^2 x)$ (note the constant here depends on the prime divisors of $x$, so we are making critical use of the fact that $x = 2^n$).  Therefore, the intersection of the two sets of sums in the above paragraph is asymptotically small, and so $x/4 - x/\log x$ is an approximation that is accurate to within a relative error of $O(1/\log^2 x)$.  Probably no further asymptotic terms can be extracted, except heuristically: the dependence on Goldbach's conjecture is pretty clear.

