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A375916 Composite numbers k == 3, 7 (mod 10) such that 5^((k-1)/2) == -1 (mod k).

%I A375916 #19 Sep 07 2024 16:03:53
%S A375916 7813,121463,195313,216457,315283,319507,353827,555397,559903,753667,
%T A375916 939727,1164083,1653667,1663213,1703677,1809697,1958503,2255843,
%U A375916 2339377,2423323,2942333,2987167,3313643,4265257,4635053,5376463,5979247,6611977,7784297,7859707
%N A375916 Composite numbers k == 3, 7 (mod 10) such that 5^((k-1)/2) == -1 (mod k).
%C A375916 Odd composite numbers k such that 5^((k-1)/2) == (5/k) = -1 (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol).
%H A375916 Jianing Song, <a href="/A375916/b375916.txt">Table of n, a(n) for n = 1..1000</a>
%e A375916 216457 is a term because 216457 = 233*929 is a composite, 216457 == 7 (mod 10), and 5^((216457-1)/2) == -1 (mod 216457).
%o A375916 (PARI) isA375916(k) = !isprime(k) && (k%10==3 || k%10==7) && Mod(5,k)^((k-1)/2) == -1
%Y A375916                                    |        b=2        |   b=3   |   b=5    |
%Y A375916 -----------------------------------+-------------------+---------+----------+
%Y A375916   (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915  |
%Y A375916 -----------------------------------+-------------------+---------+----------+
%Y A375916  (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | this seq |
%Y A375916 -----------------------------------+-------------------+---------+----------+
%Y A375916  b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | A375816  |
%Y A375916  (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
%Y A375916 -----------------------------------+-------------------+---------+----------+
%Y A375916      Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914  |
%Y A375916        (union of first two)        |                   |         |          |
%Y A375916 -----------------------------------+-------------------+---------+----------+
%Y A375916         Euler pseudoprimes         |      A006970      | A262051 | A262052  |
%Y A375916        (union of all three)        |                   |         |          |
%K A375916 nonn
%O A375916 1,1
%A A375916 _Jianing Song_, Sep 02 2024