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A289237 Find the first (maximal) string, of length exactly n, of consecutive primes that alternate between types 6*k+1 and 6*k+5 or 6*k+5 and 6*k+1. The first element is a(n).

%I A289237 #7 Jun 29 2017 10:54:57
%S A289237 53,29,67,37,449,179,5,389,89,2213,11149,10369,6761,113341,80447,
%T A289237 151909,43777,2964553,1457333,175573,809,3954889,121930481,96050953,
%U A289237 15186319,296080717,98380549,77011289,2720227693,5696814287,1572386903,4136299357,288413159
%N A289237 Find the first (maximal) string, of length exactly n, of consecutive primes that alternate between types 6*k+1 and 6*k+5 or 6*k+5 and 6*k+1. The first element is a(n).
%C A289237 By the first Formula, a(21) = 809 since A289119(21) = 809 < A289119(22).
%D A289237 R. K. Guy, Unsolved Problems in Number Theory, A4.
%H A289237 Giovanni Resta, <a href="/A289237/b289237.txt">Table of n, a(n) for n = 1..43</a>
%H A289237 Jens Kruse Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>
%F A289237 a(n) = A289119(n) if and only if n > 1 and A289119(n) < A289119(n+1).
%e A289237 {Prime[k], Mod[Prime[k], 6]} = {2, 2}, {3, 3}, {5, 5}, {7, 1}, {11, 5}, {13, 1}, {17, 5}, {19, 1}, {23, 5}, {29, 5}, {31, 1}, {37, 1}, {41, 5}, {43, 1}, {47, 5}, {53, 5}, {59, 5}, {61, 1}, {67,  1}, {71, 5}, {73, 1}, {79, 1}, . ., so a(n) = 53, 29, 67, 37 for n = 1, 2, 3, 4 and a(7) = 5.
%t A289237 i = 2; While[ Mod[ Prime[i] - Prime[i - 1], 6] != 0 || Mod[ Prime[i + 1] - Prime[i], 6] != 0, i++]; T = {Prime[i]}; Do[j = 3; While[ ! (Product[ Mod[ Prime[k + 1] - Prime[k], 6], {k, j, j + n}] != 0 && (Mod[ Prime[j] - Prime[j - 1], 6] == 0 || j == 3) && Mod[ Prime[j + n + 2] - Prime[j + n + 1], 6] == 0), j++]; T = Append[T, Prime[j]], {n, 0, 16}]; T
%Y A289237 Cf. A247384, A289118, A289119.
%K A289237 nonn
%O A289237 1,1
%A A289237 _Jonathan Sondow_, Jun 28 2017
%E A289237 a(19)-a(33) from _Giovanni Resta_, Jun 29 2017