A322796 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

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).

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.

MATHEMATICA

Array[KroneckerSymbol[6, #] &, 105, 0] (* Michael De Vlieger, Dec 31 2018 *)

Table[KroneckerSymbol[6, n], {n, 0, 100}] (* Vincenzo Librandi, Jan 01 2019 *)

PROG

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

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

CROSSREFS

Moebius transform of A035188.

Cf. A038876 (primes not inert in Q(sqrt(6))), A097934 (primes decomposing), A038877 (primes remaining inert).

Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, A188510, A011582, A316569, A011585, A289741, A011586, A109017, A011588, A390614, A388073, A388072, A011591, A011592, A011596, A011615.

Kronecker symbols {(D/n)} for positive fundamental discriminants D = 5..41: A080891, A091337, A110161, A011583, A011584, A322829, this sequence, A390615, A011587, A391502, A011589, A391503, A011590.

Sequence in context: A285530 A317542 A379275 * A109017 A110161 A134667

Adjacent sequences: A322793 A322794 A322795 * A322797 A322798 A322799

KEYWORD

sign,easy,mult

AUTHOR

Jianing Song, Dec 26 2018

EXTENSIONS

Definition corrected by Antti Karttunen, Sep 28 2019

STATUS

approved