Noticed a pattern for $n \geq 6$, the number of prime pairs $(p,q)$ where $p+q=6n$, is always greater than the pairs for $6n+2$ and $6n+4$.
For example, when $n=6$
$36$ has $8$ prime pairs
(5,31) (7,29) (13,23) (17,19) (19,17) (23,13) (29,7) (31,5)
$38$ has $3$ prime pairs
(7,31) (19,19) (31,7)
$40$ has $6$ prime pairs
(3,37) (11,29) (17,23) (23,17) (29,11) (37,3)
The charts below show peaks for $6n$ and troughs for $6n+2$ and $6n+4$
This pattern seems to continue indefinitely …
Question
Does this pattern hold true for all $6n$ peaks?
Best Answer
As a few commentors have pointed out, a prime is either $1$ or $5$ (mod $6$), and single prime has a ~$50$% of being in either group. Thus, we only need one of each to get a sum of $6n$, but we need two numbers that are equal (mod $6$) to get $6n+2$ or $6n+4$ (only a $25$% chance).
The specifics of your question have likely been worked out here. You have found a cool visual way to see this!