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A057624 Initial prime in first sequence of n primes congruent to 1 modulo 4.

%I A057624 #23 Sep 22 2024 16:10:26
%S A057624 5,13,89,389,2593,11593,11593,11593,11593,373649,766261,3358169,
%T A057624 12204889,12270077,12270077,12270077,297387757,297779117,297779117,
%U A057624 1113443017,1113443017,1113443017,1113443017,1113443017,84676452781,84676452781,689101181569,689101181569,689101181569,3278744415797,3278744415797,3278744415797,3278744415797
%N A057624 Initial prime in first sequence of n primes congruent to 1 modulo 4.
%C A057624 The sequence is infinite, by Shiu's theorem. - _Jonathan Sondow_, Jun 22 2017
%D A057624 R. K. Guy, Unsolved Problems in Number Theory, A4.
%D A057624 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 163.
%H A057624 Jens Kruse Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>.
%H A057624 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 A057624 a(9) = 11593 because "[t]his number is the first in a sequence of 9 consecutive primes all of the form 4n + 1."
%t A057624 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 ] != {1}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 4 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 19} ]
%Y A057624 Cf. A057619, A057620, A057622, A055623, A055624.
%K A057624 nonn
%O A057624 1,1
%A A057624 _Robert G. Wilson v_, Oct 09 2000
%E A057624 More terms from _Don Reble_, Nov 16 2003
%E A057624 More terms from _Jens Kruse Andersen_, May 29 2006