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%I #12 Jun 29 2017 10:55:10
%S 5,5,5,5,5,5,5,89,89,809,809,809,809,809,809,809,809,809,809,809,809,
%T 3954889,15186319,15186319,15186319,77011289,77011289,77011289,
%U 288413159,288413159,288413159,288413159,288413159,62585146739,114058236679,143014298809
%N Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 6*k+1 and 6*k+5 or 6*k+5 and 6*k+1.
%C Conjecture: the sequence is infinite. (Motivation: the string HTHTHT... of length n eventually occurs in any sufficiently long sequence of coin tosses.)
%D R. K. Guy, Unsolved Problems in Number Theory, A4.
%H Giovanni Resta, <a href="/A289119/b289119.txt">Table of n, a(n) for n = 1..43</a>
%H Jens Kruse Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>
%e For k = 3, 4, ..., 33, {Prime[k], Mod[Prime[k], 6]} = {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}, {83, 5}, {89, 5}, {97, 1}, {101, 5}, {103, 1}, {107, 5}, {109, 1}, {113, 5}, {127, 1}, {131, 5}, {137, 5}, so a(n) = 5, 5, 5, 5, 5, 5, 5, 89, 89 for n = 1, 2, ..., 9 with a(10) > 89.
%t j = 3; T = Table[ While[ Product[ Mod[ Prime[k + 1] - Prime[k], 6], {k, j, j + n}] == 0, j++]; Prime[j], {n, 0, 20}]; Prepend[T, 5]
%Y Cf. A057620, A057622, A289118.
%K nonn
%O 1,1
%A _Jonathan Sondow_, Jun 25 2017
%E a(23)-a(36) from _Giovanni Resta_, Jun 29 2017
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