Call the number of ways that even N can be expressed as the sum of two (unordered) primes g(N). g(N) tends to be much higher when N is a multiple of 3, because only a third of the "potential pairs" a+b=N are ruled out by a or b being a multiple of 3, while for other numbers two thirds are ruled out.
A consequence of that is the it's very rare to find a sequence of more than three consecutive even numbers whose g(N) is non-increasing. Here are all of them up to 10^7:
run of 4 starting at 14 run of 4 starting at 22 run of 4 starting at 910 run of 4 starting at 620620 run of 4 starting at 920920 run of 4 starting at 1521520 run of 4 starting at 1701700 run of 4 starting at 1851850 run of 4 starting at 2212210 run of 4 starting at 2662660 run of 4 starting at 2992990 run of 4 starting at 3233230 run of 4 starting at 3803800 run of 4 starting at 4344340 run of 4 starting at 4644640 run of 4 starting at 4944940 run of 4 starting at 5275270 run of 4 starting at 5515510 run of 4 starting at 5785780 run of 4 starting at 6446440 run of 4 starting at 6760390 run of 4 starting at 6806800 run of 4 starting at 7257250 run of 4 starting at 7710010 run of 4 starting at 7797790 run of 4 starting at 8128120 run of 4 starting at 8338330 run of 4 starting at 8518510 run of 4 starting at 8938930 run of 4 starting at 9202270
A suprising set of starting points! Well, not really. The repeated digits are because they are multiples of 10010. And 10010 is 2x5x7x11x13, so being a multiple of 10010 is (roughly speaking) the next best thing to being a multiple of 3.
