## S000070

First number of 6 consecutive twin prime pairs whose first numbers differ by 48 or less.

3, 1256522812841, 45183473856329, 192423825803351, 269167103025629, 552105775370291, 766403484826121, 791920568948651, 1003698437366279, 1005770184693929, 1049791325838551, 1135141716537971, 2430133231606889, 2930621484074939, 3015310184191601

1

Only the first 6-tuple (3) has first and last pair that differ by less than 48. Terms beyond the first were computed using a C++ program. The first large primes are 1256522812841 + {0, 2, 6, 8, 18, 20, 30, 32, 36, 38, 48, 50} and 45183473856329 + {0, 2, 12, 14, 18, 20, 30, 32, 42, 44, 48, 50}. Note that the second pattern is the inverse of the first.

Terms beyond the third are from Jens Kruse Andersen. - T. D. Noe, Jun 27 2014

T. D. Noe, Plot of 128 terms

T. D. Noe, Table of 128 terms

(Mma) nextTwin[p_] := Module[{p1, p2}, p1 = NextPrime[p]; While[p2 = NextPrime[p1]; p2 != p1 + 2, p1 = p2]; p1]; p = 1; tw = Table[p = nextTwin[p], {6}]; t = {}; Do[If[tw[[-1]] - tw[[1]] <= 48, AppendTo[t, tw[[1]]]]; p = nextTwin[p]; tw = Append[Rest[tw], p], {1000000}]; t

Cf. A007530 (first number of a pair of twin primes differing by 6).
Cf. S000067, S000068, S000069S000105 (3, 4, 5, and 7 twin primes).

nonn,hard,more,nice

T. D. Noe, Jun 01 2014