Let $g(n)$ be the number of ways to write $2n$ as a sum of two primes $2n=p+q$ with $p \le q$. Define $a_k$ to be the largest $n$ with $g(n)=k$. I would bet money that no-one will disprove (in the next 10 years) that $a_k$ for $10 \le k \le 19$ are $316, 346, 313, 496, 439, 454, 556, 481, 499, 706, 601$ One would expect these numbers to have no small odd divisors (and each is a power of 2 times a prime except 499=13*37) but I would expect them to be a about evenly split between $1 \mod 3$ and $2 \mod 3$. Yet all of these are $1 \mod 3$. Is there a model which accounts for this?

Discussion: of course we do not know if $a_k$ is well defined although we suspect it is. Data exists for $g(n)$, For example [decompositions of 2n into an unordered sum of two odd primes][1] links to a table for $1 \le n \le 20000$) and the numbers have been computed up to $2.5 \cdot 10^8$. The estimates are based on the size of n and the set of odd primes dividing it. This would predicit that from some point on we never have $g(n+m)<g(n)$ with $n+m$ a multiple of 3 but $n$ not a multiple of 3. That seems to be true (I think).

Question: Is there any reason to expect that congruence class mod 3 is correlated with g(n)? Has this been observed [1]: http://oeis.org/A002375