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A117889 Period of sequence {Kronecker(-n,k), k = 1, 2, 3, ...}, or 0 if the sequence is not periodic. 2
0, 8, 3, 4, 0, 24, 7, 8, 0, 40, 11, 6, 0, 56, 15, 4, 0, 24, 19, 20, 0, 88, 23, 24, 0, 104, 3, 14, 0, 120, 31, 8, 0, 136, 35, 12, 0, 152, 39, 40, 0, 168, 43, 22, 0, 184, 47, 6, 0, 40, 51, 52, 0, 24, 55, 56, 0, 232, 59, 30, 0, 248, 21, 4, 0, 264, 67, 68, 0, 280, 71, 24, 0, 296, 15, 38 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Jianing Song, Nov 24 2018: (Start)

The sequence {Kronecker(-n,k)} forms a Dirichlet character modulo n if and only if n == 0, 3 (mod 4).

Let n = 2^t*s, s odd, then a(n) = A117888(n) if and only if t is odd, a(n) = A302138(n) if and only if t is odd or s == 3 (mod 4) (or both). (End)

LINKS

Table of n, a(n) for n=1..76.

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

FORMULA

Let n = 2^t*s, s odd, then a(n) = 4*A007947(n) if t is odd; A007947(n) if t is even and s == 3 (mod 4); 2*A007947(n) if t is even and t > 0 and s == 1 (mod 4); 0 if t = 0 and s == 1 (mod 4). - Jianing Song, Nov 24 2018

MATHEMATICA

per[lst_] := FindTransientRepeat[lst, 4] // Last // Length;

a[n_] := per[Table[KroneckerSymbol[-n, k], {k, 1, 1200}]];

Array[a, 76] (* Jean-Fran├žois Alcover, Oct 08 2018 *)

CROSSREFS

Cf. A007947.

Cf. A117888 (period of Kronecker(n,k)), A302138 (period of Kronecker(k,n)).

Sequence in context: A199863 A181180 A271521 * A021927 A145594 A194731

Adjacent sequences:  A117886 A117887 A117888 * A117890 A117891 A117892

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Mar 30 2006

EXTENSIONS

Edited by N. J. A. Sloane, May 31 2009

STATUS

approved

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Last modified October 24 20:35 EDT 2021. Contains 348233 sequences. (Running on oeis4.)