First number of 3 consecutive twin prime pairs whose first numbers differ by 18 or less.
3, 5, 11, 179, 809, 5639, 9419, 18041, 62969, 88799, 97841, 109829, 165701, 284729, 392261, 452519, 626609, 663569, 663581, 855719, 983429, 1002341, 1003349, 1068701, 1146779, 1155611, 1322159, 1329701, 1592861, 1678751, 1718861, 1748471, 2116559, 2144489
1
Only the first two triples have first and last terms that differ by less than 18. The next triple of twin primes is 11 + {0, 2, 6, 8, 18, 20}. After the term 5, all sets of 3 twin primes have one of two forms: p + {0, 2, 6, 8, 18, 20} or p + {0, 2, 12, 14, 18, 20} 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], {3}]; t = {}; Do[If[tw[[-1]] - tw[[1]] <= 18, 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. S000068, S000069, S000070, S000105 (4 to 7 twin primes).
nonn,nice
T. D. Noe, Jun 01 2014