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A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.

%I A023347 #26 Jul 14 2025 06:20:20
%S A023347 831167,1154567,2502767,3019787,3675197,5056577,6352487,14519177,
%T A023347 26724377,43003577,47378927,47695607,56406197,86332457,86611757,
%U A023347 99568757,121967987,126435527,127990997,128149127,128975057,145281557,155715407
%N A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.
%H A023347 John Cerkan, <a href="/A023347/b023347.txt">Table of n, a(n) for n = 1..10000</a>
%F A023347 {p, 8p+1, 64p+9, 512p+73, 4096p+585, 32768p+4681} are all primes, where the initial p is prime.
%F A023347 a(n) == 197 (mod 210). - _John Cerkan_, Nov 04 2016
%e A023347 First chain is {831167, 6649337, 53194697, 425557577, 3404460617, 27235684937};
%e A023347 If p is congruent to {1,3,7,9} mod 10, then consecutive iterates are congruent to {9,5,7,3}, {3,1,7,5}, {5,9,7,1} respectively; so only 10k+7 may remain prime through five iterations, as sequence demonstrates nicely. - _Labos Elemer_, Jul 23 2003
%t A023347 k=0; m=8; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
%t A023347 it5Q[n_]:=AllTrue[Rest[NestList[8#+1&,n,5]],PrimeQ]; Select[Prime[Range[ 9*10^6]],it5Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Sep 12 2014 *)
%Y A023347 Cf. A085956, A086361, A086362, A023330, A059766, A023287, A000668, A076481, A086127.
%Y A023347 Subsequence of A005123, A023228, A023260, A023291, and A023319.
%K A023347 nonn
%O A023347 1,1
%A A023347 _David W. Wilson_