A322796 a(n) = Kronecker symbol (6/n).

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

Offset: 0

Author

Jianing Song, Dec 26 2018

Keywords

• sign
• easy
• mult

Comments

Period 24: repeat [0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1].

Also a(n) = Kronecker symbol (24/n).

This sequence is one of the seven non-principal real Dirichlet characters modulo 24. The other six are Jacobi or Kronecker symbols {(-6/n)} (or {(n/6)}, {(-24/n)}, {(n/24)}, A109017), {(-12/n)} (or {(n/12)}, A134667), {(12/n)} (A110161), {(-18/n)} (or {(-72/n)}), {(18/n)} (or {(72/n)}, {(n/72)}) and {(-36/n)}. These sequences all become the same after taking absolute values.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65543
• Eric Weisstein's World of Mathematics, Kronecker Symbol (contains this sequence)
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,-1).

Crossrefs

Cf. A035188 (inverse Moebius transform).

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), this sequence (d=24).

Programs

Magma

[KroneckerSymbol(6, n): n in [0..100]]; // Vincenzo Librandi, Jan 01 2019

Mathematica

Array[KroneckerSymbol[6, #] &, 105, 0] (* Michael De Vlieger, Dec 31 2018 *)
Table[KroneckerSymbol[6, n], {n, 0, 100}] (* Vincenzo Librandi, Jan 01 2019 *)

PARI

a(n) = kronecker(6, n); \\ --- Argument order corrected by Antti Karttunen, Sep 27 2019

Formula

a(n) = 1 for n == 1, 5, 19, 23 (mod 24); -1 for n == 7, 11, 13, 17 (mod 24); 0 for n that are not coprime with 21.

Completely multiplicative with a(p) = a(p mod 24) for primes p.

a(n) = A091337(n)*A102283(n).

a(n) = A109017(n+12) = A109017(n-12).

a(n) = a(-n) = a(n+24) for all n in Z.

Extensions

Definition corrected by Antti Karttunen, Sep 28 2019