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A375918 Composite numbers k == 5, 7 (mod 12) such that 3^((k-1)/2) == -1 (mod k).

%I A375918 #18 Sep 07 2024 16:03:24
%S A375918 703,1891,3281,8911,12403,16531,44287,63139,79003,97567,105163,152551,
%T A375918 182527,188191,211411,218791,288163,313447,320167,364231,385003,
%U A375918 432821,453259,497503,563347,638731,655051,658711,801139,859951,867043,973241,994507,1024651,1097227
%N A375918 Composite numbers k == 5, 7 (mod 12) such that 3^((k-1)/2) == -1 (mod k).
%C A375918 Odd composite numbers k such that 3^((k-1)/2) == (3/k) = -1 (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol).
%H A375918 Jianing Song, <a href="/A375918/b375918.txt">Table of n, a(n) for n = 1..1000</a>
%e A375918 3281 is a term because 3281 = 17*193 is composite, 3281 == 5 (mod 12), and 3^((3281-1)/2) == -1 (mod 3281).
%o A375918 (PARI) isA375918(k) = !isprime(k) && (k%12==5 || k%12==7) && Mod(3,k)^((k-1)/2) == -1
%Y A375918                                    |        b=2        |   b=3    |   b=5   |
%Y A375918 -----------------------------------+-------------------+----------+---------+
%Y A375918   (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917  | A375915 |
%Y A375918 -----------------------------------+-------------------+----------+---------+
%Y A375918  (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | this seq | A375916 |
%Y A375918 -----------------------------------+-------------------+----------+---------+
%Y A375918  b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490  | A375816 |
%Y A375918  (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |          |         |
%Y A375918 -----------------------------------+-------------------+----------+---------+
%Y A375918      Euler-Jacobi pseudoprimes     |      A047713      | A048950  | A375914 |
%Y A375918        (union of first two)        |                   |          |         |
%Y A375918 -----------------------------------+-------------------+----------+---------+
%Y A375918         Euler pseudoprimes         |      A006970      | A262051  | A262052 |
%Y A375918        (union of all three)        |                   |          |         |
%K A375918 nonn
%O A375918 1,1
%A A375918 _Jianing Song_, Sep 02 2024