Least twin prime p1 such that there is a smaller twin prime p2 with p1 - p2 = 2*n.

5, 7, 11, 11, 13, 17, 17, 19, 29, 31, 29, 29, 29, 31, 41, 43, 41, 41, 41, 43, 59, 61, 59, 59, 61, 59, 59, 59, 61, 71, 73, 71, 71, 71, 73, 101, 103, 107, 107, 109, 101, 101, 103, 101, 101, 103, 101, 101, 101, 103, 107, 107, 109, 137, 139, 311, 311, 313, 137

1

It appears that such twin primes occur for all n. Moreover, it appears that there are an infinite number of twin primes that differ by a given even number. The prime p2 is given in S000946.

T. D. Noe, Plot of 10000 terms

T. D. Noe, Table of 10000 terms

Wikipedia, Twin prime

(Mma) prms = 100; tp = Select[Prime[Range[2, prms]], PrimeQ[# + 2] &]; tp = Union[tp, tp + 2]; u = Union[Select[Flatten[mat = Table[a - b, {a, tp}, {b, tp}]], # > 0 &]]; d = Differences[u]; mx = Position[d, _?(# > 2 &), 1, 1][[1, 1]]; nn = u[[mx]]/2; t = Table[0, {nn}]; n = 1; cnt = 0; While[cnt < nn, n++; diff = Reverse[Table[tp[[n]] - tp[[i]], {i, n - 1}]/2]; Do[If[diff[[i]] <= nn && t[[diff[[i]]]] == 0, cnt++; t[[diff[[i]]]] = tp[[n]]], {i, n - 1}]]; t

Cf. A129363 (partitions of 2n into the sum of two twin primes), S000946.

nonn,nice

T. D. Noe, Sep 29 2016