cp's OEIS Frontend

A375816 Odd numbers k > 1 such that gcd(5,k) = 1 and 5^((k-1)/2) == -(5/k) (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 5 (A262052) that are not Euler-Jacobi pseudoprimes to base 5 (A375914).

%I A375816 #38 Sep 07 2024 16:03:45
%S A375816 217,13333,16297,23653,30673,44173,46657,48133,56033,98173,130417,
%T A375816 131977,136137,179893,188113,190513,197633,267977,334153,334657,
%U A375816 347777,360533,407353,412933,421637,486157,667153,670033,677917,694153,710533,765073,839833,935137,997633
%N A375816 Odd numbers k > 1 such that gcd(5,k) = 1 and 5^((k-1)/2) == -(5/k) (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 5 (A262052) that are not Euler-Jacobi pseudoprimes to base 5 (A375914).
%C A375816 Note that if k is odd and b^((k-1)/2) == -(b/k) (mod k), then taking Jacobi symbol modulo k (which depends only on the remainder modulo k) yields (b/k)^((k-1)/2) = -(b/k), or (b/k)^((k+1)/2) = -1. This implies that (k+1)/2 is odd, so k == 1 (mod 4). Moreover, if k > 1, then (b/k) = -1 (see the Math Stack Exchange link below), so b^((k-1)/2) == 1 (mod k). In particular, this sequence is equivalent to "numbers k == 13, 17 (mod 20) such that 5^((k-1)/2) == 1 (mod k)". [Comment rewritten by _Jianing Song_, Sep 07 2024]
%H A375816 Jianing Song, <a href="/A375816/b375816.txt">Table of n, a(n) for n = 1..1000</a>
%H A375816 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/4966167">There are no a in Z and odd k > 1 such that (a/k) = 1 and a^((k-1)/2) == -1 (mod k)</a>
%e A375816 217 is a term because (5/217) = -1, and 5^((217-1)/2) == 1 (mod 217).
%o A375816 (PARI) isA375816(k) = (k>1) && gcd(k,10)==1 && Mod(5,k)^((k-1)/2)==-kronecker(5,k)
%o A375816 (PARI) isA375816(k) = (k%20==13 || k%20==17) && Mod(5,k)^((k-1)/2)==1
%Y A375816                                    |        b=2        |   b=3   |   b=5    |
%Y A375816 -----------------------------------+-------------------+---------+----------+
%Y A375816   (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915  |
%Y A375816 -----------------------------------+-------------------+---------+----------+
%Y A375816  (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916  |
%Y A375816 -----------------------------------+-------------------+---------+----------+
%Y A375816  b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | this seq |
%Y A375816  (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
%Y A375816 -----------------------------------+-------------------+---------+----------+
%Y A375816      Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914  |
%Y A375816        (union of first two)        |                   |         |          |
%Y A375816 -----------------------------------+-------------------+---------+----------+
%Y A375816         Euler pseudoprimes         |      A006970      | A262051 | A262052  |
%Y A375816        (union of all three)        |                   |         |          |
%K A375816 nonn
%O A375816 1,1
%A A375816 _Jianing Song_, Sep 01 2024