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A214151 Numbers k from the set == 5 (mod 6) with the property that 3^((3*k-1)/2) == 3 (mod k) and 2^((k-1)/2) == (k-1) (mod k).

%I A214151 #34 Dec 12 2023 10:38:38
%S A214151 11,59,83,107,131,179,227,251,347,419,443,467,491,563,587,659,683,827,
%T A214151 947,971,1019,1091,1163,1187,1259,1283,1307,1427,1451,1499,1523,1571,
%U A214151 1619,1667,1787,1811,1907,1931,1979,2003,2027,2099,2243,2267
%N A214151 Numbers k from the set == 5 (mod 6) with the property that 3^((3*k-1)/2) == 3 (mod k) and 2^((k-1)/2) == (k-1) (mod k).
%C A214151 All composites in this sequence are 2-pseudoprimes, see A001567, and strong pseudoprimes to base 2, A001262.
%C A214151 The subsequence of these composites begins: 1441091, 3587553971, 4528686251, 23260036451, 47535120323, 61070250323, 90474845819, 143193768587, 162016315907, 173868807611, 180998962187, 238364070323, 285370693931, 298577370323, ...
%C A214151 Perhaps this sequence contains all the terms of the sequence A107007 or A168539.
%H A214151 Harvey P. Dale, <a href="/A214151/b214151.txt">Table of n, a(n) for n = 1..1000</a>
%p A214151 isA214151 := proc(n)
%p A214151     if (n mod 6 = 5) and modp(2 &^ ((n-1)/2),n)  = n-1 and modp(3 &^ ((3*n-1)/2),n)  = 3 then
%p A214151         true;
%p A214151     else
%p A214151         false;
%p A214151     end if;
%p A214151 end proc:
%p A214151 for n from 5 by 6 do
%p A214151     if isA214151(n) then
%p A214151         print(n) ;
%p A214151     end if;
%p A214151 end do: # _R. J. Mathar_, Jul 20 2012
%t A214151 Select[Range[5,2500,6],PowerMod[3,(3#-1)/2,#]==3&&PowerMod[2,(#-1)/2,#] == #-1&] (* _Harvey P. Dale_, Mar 14 2022 *)
%o A214151 (PARI) for(n=0, 200, b=6*n+5; if(Mod(3, b)^((3*b-1)/2)==3, if(Mod(2, b)^((b-1)/2)==b-1 , print1(b, ", "))));
%Y A214151 Subsequence of A176997.
%Y A214151 Cf. A003629, A006970, A175865.
%K A214151 nonn
%O A214151 1,1
%A A214151 _Alzhekeyev Ascar M_, Jul 05 2012