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%I #21 Jul 02 2017 14:54:30
%S 3,3,3,23,47,131,131,233,233,521,521,521,521,521,521,51749,505049,
%T 1391087,2264839,2556713,2569529,2569529,6160043,6160043,6160043,
%U 43679609,43679609,198572029,701575297,5552898499,6639843979,9005520203,9005520203,99052377023
%N Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 4*k+1 and 4*k+3 or 4*k+3 and 4*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="/A289118/b289118.txt">Table of n, a(n) for n = 1..45</a>
%H Jens Kruse Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>
%F a(n) = A247384(n) if and only if n > 1 and a(n) < a(n+1).
%e {Prime[k], Mod[ Prime[k], 4]} = {{3, 3}, {5, 1}, {7, 3}, {11, 3}, {13, 1}, {17, 1}, {19, 3}, {23, 3}, {29, 1}}, {31, 3}, {37, 1}, . . for k = 2, 3, 4, . ., so a(n) = 3, 3, 3, 23 for n = 1, 2, 3, 4.
%t j = 2; T = Table[ While[ Product[ Mod[ Prime[k + 1] - Prime[k], 4], {k, j, j + n}] == 0, j++]; Prime[j], {n, 0, 15}]; Prepend[T, 3]
%Y For the least prime at the start of such a string of length exactly n, see A247384.
%Y Cf. A098058, A289119.
%K nonn
%O 1,1
%A _Jonathan Sondow_, Jun 25 2017
%E a(18)-a(27) from _Alois P. Heinz_, Jun 26 2017
%E a(28)-a(34) from _Giovanni Resta_, Jul 02 2017
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