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A306310 Odd numbers k > 1 such that 2^((k-1)/2) == -(2/k) = -A091337(k) (mod k), where (2/k) is the Jacobi (or Kronecker) symbol.

%I A306310 #44 Sep 07 2024 16:03:20
%S A306310 341,5461,10261,15709,31621,49981,65077,83333,137149,176149,194221,
%T A306310 215749,219781,276013,282133,534061,587861,611701,653333,657901,
%U A306310 665333,688213,710533,722261,738541,742813,769757,950797,1064053,1073021,1109461,1141141,1357621,1398101
%N A306310 Odd numbers k > 1 such that 2^((k-1)/2) == -(2/k) = -A091337(k) (mod k), where (2/k) is the Jacobi (or Kronecker) symbol.
%C A306310 All terms are composite because for odd primes p we always have 2^((p-1)/2) == (2/p) = A091337(p) (mod p).
%C A306310 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 8) such that 2^((k-1)/2) == 1 (mod k)". [Comment rewritten by _Jianing Song_, Sep 07 2024]
%C A306310 Also numbers k in A001567 and congruent to 5 modulo 8 such that k - 1 divided by the multiplicative order of 2 modulo k is an even number.
%C A306310 Euler pseudoprimes (A006970) that are not Euler-Jacobi pseudoprimes (A047713). - _Amiram Eldar_, Oct 28 2019
%H A306310 Amiram Eldar, <a href="/A306310/b306310.txt">Table of n, a(n) for n = 1..1000</a>
%H A306310 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 A306310 341 is a term because (2/341) = -1, and 2^((341-1)/2) == 1 (mod 341).
%o A306310 (PARI) isA306310(k)=(k%8==5) && Mod(2, k)^((k-1)/2)==1
%o A306310 (PARI) isok(k) = (k>1) && (k%2) && (Mod(2, k)^((k-1)/2) == Mod(-kronecker(2, k), k)); \\ _Michel Marcus_, Feb 07 2019
%Y A306310 Cf. A091337, A001567, A007814.
%Y A306310                                    |        b=2        |   b=3   |   b=5   |
%Y A306310 -----------------------------------+-------------------+---------+---------+
%Y A306310   (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915 |
%Y A306310 -----------------------------------+-------------------+---------+---------+
%Y A306310  (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916 |
%Y A306310 -----------------------------------+-------------------+---------+---------+
%Y A306310  b^((k-1)/2)==-(b/k) (mod k), also |      this seq     | A375490 | A375816 |
%Y A306310  (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |         |
%Y A306310 -----------------------------------+-------------------+---------+---------+
%Y A306310      Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914 |
%Y A306310        (union of first two)        |                   |         |         |
%Y A306310 -----------------------------------+-------------------+---------+---------+
%Y A306310         Euler pseudoprimes         |      A006970      | A262051 | A262052 |
%Y A306310        (union of all three)        |                   |         |         |
%K A306310 nonn
%O A306310 1,1
%A A306310 _Jianing Song_, Feb 06 2019