If 3|N, and there are 2*k consecutive primes occurring symmetrically around N, k on each side, there can be up to k - 1 or whatever, of gaps with non-zero residues (mod 3) (between consecutive primes, except the first gap of p - N or N - p) on either side. But they have to occur in a specific sequence. The non-zero residues of gaps can be dispersed among any number of gaps of length divisible by 3, but on one side have to occur in the sequence

1,1,2,1,2,1,2,1,... (the 2,1 repeats)

and the nonzero residues (mod 3) of the corresponding gaps on the other side of N must occur in the sequence

2,2,1,2,1,2,1,2,... (the 1,2 repeats)

The number of possibilities obviously increases without bound as k increases without bound.

I'm too lazy to work out how fast. Either that, or I'm just not very good at this sort of thing