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).

1541, 2465, 4961, 30857, 31697, 72041, 83333, 162401, 192713, 206981, 258017, 359369, 544541, 565001, 574397, 653333, 929633, 1018601, 1032533, 1133441, 1351601, 1373633, 1904033, 1953281, 2035661, 2797349, 2864501, 3264797, 3375041, 3554633, 3562361, 3636161

Offset

1,1

Comments

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]

Links

Jianing Song, Table of n, a(n) for n = 1..1000

Mathematics Stack Exchange, There are no a in Z and odd k > 1 such that (a/k) = 1 and a^((k-1)/2) == -1 (mod k).

Example

1541 is a term because (3/1541) = -1, and 3^((1541-1)/2) == 1 (mod 1541).

Mathematica

q[k_] := Mod[k, 12] == 5 && PowerMod[3, (k-1)/2, k] == 1; Select[Range[10^6], q] (* Amiram Eldar, Mar 21 2026 *)

Prog

(PARI) isA375490(k) = (k>1) && gcd(k, 6)==1 && Mod(3, k)^((k-1)/2)==-kronecker(3, k)
(PARI) isA375490(k) = k%12==5 && Mod(3, k)^((k-1)/2)==1 \\ Jianing Song, Sep 07 2024

Crossrefs

For a list of sequences related to Euler-Jacobi pseudoprimes and Euler pseudoprimes, see A306310.

Sequence in context: A092717 A203134 A083734 * A137598 A252966 A133560

Adjacent sequences: A375487 A375488 A375489 * A375491 A375492 A375493

Keyword

nonn

Author

Jianing Song, Sep 01 2024

Status

approved