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%I #23 Oct 19 2017 03:13:52
%S 3,7,199,199,463,463,463,36551,39607,183091,241603,241603,241603,
%T 9177431,9177431,95949311,105639091,341118307,727334879,727334879,
%U 1786054147,1786054147,22964264027,54870713243,79263248027,113391385603
%N Initial prime in first sequence of n primes congruent to 3 modulo 4.
%C The sequence is infinite, by Shiu's theorem. - _Jonathan Sondow_, Jun 22 2017
%D R. K. Guy, "Unsolved Problems in Number Theory", A4
%H Giovanni Resta, <a href="/A057619/b057619.txt">Table of n, a(n) for n = 1..36</a> (terms < 4*10^14)
%H J. K. Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>.
%H D. K. L. Shiu, <a href="http://dx.doi.org/10.1112/S0024610799007863">Strings of Congruent Primes</a>, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [<a href="http://www.ams.org/mathscinet-getitem?mr=1760689">MR1760689</a>]
%e a(13) = 241603 because this number is the first in a sequence of 13 consecutive primes all of the form 4n + 3.
%t NextPrime[ n_Integer ] := Module[ {k = n + 1}, While[ ! PrimeQ[ k ], k++ ]; Return[ k ] ]; PrevPrime[ n_Integer ] := Module[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; Return[ k ] ]; p = 0; Do[ a = Table[ -1, {n} ]; k = Max[ 1, p ]; While[ Union[ a ] != {3}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 4 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 18} ]
%t With[{prs=Table[If[Mod[Prime[n],4]==3,1,0],{n,4646*10^6}]},Prime/@ Table[ SequencePosition[prs,PadRight[{},k,1],1][[1,1]],{k,26}]] (* The program will take a long time to run and requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Sep 28 2017 *)
%Y Cf. A057620, A057622, A057624.
%K nonn
%O 1,1
%A _Robert G. Wilson v_, Oct 09 2000
%E More terms from _Don Reble_, Nov 16 2003
%E More terms from _Jens Kruse Andersen_, May 29 2006
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