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A375917 Composite numbers k == 1, 11 (mod 12) such that 3^((k-1)/2) == 1 (mod k).

%I A375917 #18 Sep 07 2024 16:03:28
%S A375917 121,1729,2821,7381,8401,10585,15457,15841,18721,19345,23521,24661,
%T A375917 28009,29341,31621,41041,46657,47197,49141,50881,52633,55969,63973,
%U A375917 74593,75361,82513,87913,88573,93961,111361,112141,115921,125665,126217,138481,148417,172081
%N A375917 Composite numbers k == 1, 11 (mod 12) such that 3^((k-1)/2) == 1 (mod k).
%C A375917 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).
%C A375917 It seems that most terms are congruent to 1 modulo 12. The first terms congruent to 11 modulo 12 are 1683683, 1898999, 2586083, 2795519, 4042403, 4099439, 5087171, 8243111, ...
%H A375917 Jianing Song, <a href="/A375917/b375917.txt">Table of n, a(n) for n = 1..1000</a>
%e A375917 1683683 is a term because 1683683 = 59*28537 is composite, 1683683 == 11 (mod 12), and 3^((1683683-1)/2) == 1 (mod 1683683).
%o A375917 (PARI) isA375917(k) = (k>1) && !isprime(k) && (k%12==1 || k%12==11) && Mod(3,k)^((k-1)/2) == 1
%Y A375917                                    |        b=2        |   b=3    |   b=5   |
%Y A375917 -----------------------------------+-------------------+----------+---------+
%Y A375917   (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | this seq | A375915 |
%Y A375917 -----------------------------------+-------------------+----------+---------+
%Y A375917  (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918  | A375916 |
%Y A375917 -----------------------------------+-------------------+----------+---------+
%Y A375917  b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490  | A375816 |
%Y A375917  (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |          |         |
%Y A375917 -----------------------------------+-------------------+----------+---------+
%Y A375917      Euler-Jacobi pseudoprimes     |      A047713      | A048950  | A375914 |
%Y A375917        (union of first two)        |                   |          |         |
%Y A375917 -----------------------------------+-------------------+----------+---------+
%Y A375917         Euler pseudoprimes         |      A006970      | A262051  | A262052 |
%Y A375917        (union of all three)        |                   |          |         |
%K A375917 nonn
%O A375917 1,1
%A A375917 _Jianing Song_, Sep 02 2024