## S000068

First number of 4 consecutive twin prime pairs whose first numbers differ by 30 or less.

3, 5, 11, 626597, 663569, 1322147, 2144477, 2668217, 6510191, 6937937, 10187909, 11495579, 11721767, 18873497, 18873509, 25658429, 39659531, 39851291, 46533467, 80572157, 84099317, 86944601, 91814711, 93956099, 123911531, 128469149, 129902021, 148979837

1

Only the first two quadruples have first and last terms that differ by less than 30. The next quadruple of twin primes is  11 + {0, 2, 6, 8, 18, 20, 30, 32}. After the term 5, all sets of 4 twin primes have one of three forms: p + {0, 2, 6, 8, 18, 20, 30, 32}, p + {0, 2, 12, 14, 18, 20, 30, 32}, or p + {0, 2, 12, 14, 24, 26, 30, 32} for some prime p.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 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], {4}]; t = {}; Do[If[tw[[-1]] - tw[[1]] <= 30, 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, S000069, S000070S000105 (3, 5, 6, and 7 twin primes).

nonn,nice

T. D. Noe, Jun 01 2014