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Initial prime in first sequence of n primes congruent to 1 modulo 8.
0

%I #14 Sep 29 2024 11:32:33

%S 17,89,2593,20809,208393,2663897,7336457,42453937,42453937,1506473153,

%T 24771906961,123737745289,152368449001,152368449001,4990160038937,

%U 50515057659673,169068296123497,402384411007849

%N Initial prime in first sequence of n primes congruent to 1 modulo 8.

%H J. K. Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>.

%e a(4) = 20809 because this number is the first in a sequence of 4 consecutive primes all of the form 8n + 1.

%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 ] != {1}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 8 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 9} ] a(10) > 123700000.

%t Module[{nn=75100000,pr8},pr8=Table[If[Mod[p,8]==1,1,0],{p,Prime[Range[nn]]}];Prime[#]&/@Table[SequencePosition[ pr8,PadRight[ {},n,1],1],{n,10}]][[;;,1,1]] (* The program generates the first 10 terms of the sequence. *) (* _Harvey P. Dale_, Sep 29 2024 *)

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Oct 10 2000

%E More terms from _Jens Kruse Andersen_, May 28 2006

%E a(16)-a(18) from _Giovanni Resta_, Aug 04 2013