A117888 Period of sequence {Kronecker(n,k), k = 1, 2, 3, ...}, or 0 if the sequence is not periodic.

1, 8, 0, 2, 5, 24, 0, 8, 3, 40, 0, 12, 13, 56, 0, 2, 17, 24, 0, 10, 21, 88, 0, 24, 5, 104, 0, 28, 29, 120, 0, 8, 33, 136, 0, 6, 37, 152, 0, 40, 41, 168, 0, 44, 15, 184, 0, 12, 7, 40, 0, 26, 53, 24, 0, 56, 57, 232, 0, 60, 61, 248, 0, 2, 65, 264, 0, 34, 69, 280, 0, 24, 73, 296, 0, 76

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, 1 (mod 4).

Let n = 2^t*s, s odd, then a(n) = A117889(n) if and only if t is odd, a(n) = A302138(n) if and only if t is odd or s == 1 (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 == 1 (mod 4); 2*A007947(n) if t is even and t > 0 and s == 3 (mod 4); 0 if t = 0 and s == 3 (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-François Alcover, Oct 08 2018 *)

Crossrefs

Cf. A007947.

Cf. A117889 (period of Kronecker(-n,k)), A302138 (period of Kronecker(k,n)).

Sequence in context: A342200 A265294 A062522 * A331542 A295330 A094240

Adjacent sequences: A117885 A117886 A117887 * A117889 A117890 A117891

Keyword

nonn

Author

Eric W. Weisstein, Mar 30 2006

Extensions

Definition corrected by N. J. A. Sloane, May 31 2009

Status

approved