I have a conjecture related to the strong Goldbach conjecture and the Goldbach function. It is that: for any $g(E)$, there are a finite number of even numbers which can be expressed as a sum of two primes in precisely that many ways. More formally, I'm introducing a new function $h(g(E))$, which is defined as the number of evens which yield that $g(E)$. My conjecture is that, for any finite $n$, $h(n)$ is finite.
The strong Goldbach conjecture is simply that $h(0) = 2$ (only 2 and 4 cannot be expressed as the sum of odd primes).
My conjecture seems to me very likely to be true. The question then arises: what does the function $h(x)$ look like. Are there any good approximations?