cp's OEIS Frontend

A375490 Odd numbers k > 1 such that gcd(3,k) = 1 and 3^((k-1)/2) == -(3/k) (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 3 (A262051) that are not Euler-Jacobi pseudoprimes to base 3 (A048950).

%I A375490 #39 Sep 07 2024 16:03:16
%S A375490 1541,2465,4961,30857,31697,72041,83333,162401,192713,206981,258017,
%T A375490 359369,544541,565001,574397,653333,929633,1018601,1032533,1133441,
%U A375490 1351601,1373633,1904033,1953281,2035661,2797349,2864501,3264797,3375041,3554633,3562361,3636161
%N A375490 Odd numbers k > 1 such that gcd(3,k) = 1 and 3^((k-1)/2) == -(3/k) (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 3 (A262051) that are not Euler-Jacobi pseudoprimes to base 3 (A048950).
%C A375490 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 == 5 (mod 12) such that 3^((k-1)/2) == 1 (mod k)". [Comment rewritten by _Jianing Song_, Sep 07 2024]
%H A375490 Jianing Song, <a href="/A375490/b375490.txt">Table of n, a(n) for n = 1..1000</a>
%H A375490 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 A375490 1541 is a term because (3/1541) = -1, and 3^((1541-1)/2) == 1 (mod 1541).
%o A375490 (PARI) isA375490(k) = (k>1) && gcd(k,6)==1 && Mod(3,k)^((k-1)/2)==-kronecker(3,k)
%o A375490 (PARI) isA375490(k) = k%12==5 && Mod(3,k)^((k-1)/2)==1 \\ _Jianing Song_, Sep 07 2024
%Y A375490                                    |        b=2        |   b=3    |   b=5   |
%Y A375490 -----------------------------------+-------------------+----------+---------+
%Y A375490   (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917  | A375915 |
%Y A375490 -----------------------------------+-------------------+----------+---------+
%Y A375490  (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918  | A375916 |
%Y A375490 -----------------------------------+-------------------+----------+---------+
%Y A375490  b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | this seq | A375816 |
%Y A375490  (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |          |         |
%Y A375490 -----------------------------------+-------------------+----------+---------+
%Y A375490      Euler-Jacobi pseudoprimes     |      A047713      | A048950  | A375914 |
%Y A375490        (union of first two)        |                   |          |         |
%Y A375490 -----------------------------------+-------------------+----------+---------+
%Y A375490         Euler pseudoprimes         |      A006970      | A262051  | A262052 |
%Y A375490        (union of all three)        |                   |          |         |
%K A375490 nonn
%O A375490 1,1
%A A375490 _Jianing Song_, Sep 01 2024