A091338 a(n) = (3/n), where (k/n) is the Kronecker symbol.

1, -1, 0, 1, -1, 0, -1, -1, 0, 1, 1, 0, 1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, -1, 0, -1, -1, 0, -1, -1, 0, 1, 1, 0, 1, 1, 0, 1, -1, 0, -1, 1, 0, -1, 1, 0, 1, -1, 0, 1, -1, 0, -1, 1, 0, 1, 1, 0, 1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, -1, 0, -1, -1, 0, -1, -1, 0, 1, 1, 0, 1, 1, 0, -1, -1, 0, -1, 1, 0, -1, 1, 0, 1, -1, 0, 1, -1, 0

Offset: 1

Author

Eric W. Weisstein, Dec 30 2003

Keywords

• sign
• mult

Comments

a(2n+1) has period 6, i.e., if n == 1 (mod 2) then a(n+12) = a(n). A.H.M. Smeets, Jan 23 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Jean-Paul Allouche, Leo Goldmakher, Mock characters and the Kronecker symbol, arXiv:1608.03957 [math.NT], 2016.
• Eric Weisstein's World of Mathematics, Kronecker Symbol

Programs

Magma

[KroneckerSymbol(3,n): n in [1..100]]; // Vincenzo Librandi, Aug 16 2016

Maple

A091338 := proc(n)
        numtheory[jacobi](3,n) ;
end proc: # R. J. Mathar, Nov 03 2011

Mathematica

Table[KroneckerSymbol[3, n], {n, 1, 100}] (* Vincenzo Librandi, Aug 16 2016 *)

PARI

a(n)=kronecker(3,n)

Formula

If n==0 (mod 3) a(n)=0; for p ==1 or 11 (mod 12) (i.e., p>3 in A038874), a(p)=+1; for p==2, 5 or 7 (mod 12) (i.e., p in A038875), a(p)=-1. - Benoit Cloitre, Jan 03 2004

From A.H.M. Smeets, Aug 01 2018: (Start)

Conjecture:

a(n) = 0 if and only if (n mod 3 = 0),

a(n) = 1 if (n mod 12 = 1 or n mod 12 = 11 or n mod 48 = 4 or n mod 48 = 44),

a(n) = -1 if (n mod 12 = 5 or n mod 12 = 7 or n mod 48 = 20 or n mod 48 = 28),

a(2) = -1, a(12*n+10) = -a(12*n+2) and a(12*n+14) = a(12*n+10) for n >= 0,

a(24*n+8) = -a(12*n+4) and a(24*n+16) = -a(12*n+4) for n >= 0. (End)

From A.H.M. Smeets, Aug 01 2018: (Start)

a(2*n+1) = 1 if and only if (n mod 6 = 0 or n mod 6 = 5),

a(2*n+1) = -1 if and only if (n mod 6 = 2 or n mod 6 = 3),

a(2*n+1) = 0 if and only if n mod 3 = 1,

a(2*n) = -a(n). (End)

Extensions

More terms from Benoit Cloitre, Jan 03 2004