A226162 a(n) = Kronecker Symbol (-5/n), n >= 0.

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

Offset: 0

Author

Wolfdieter Lang, May 29 2013

Keywords

• sign
• mult

Comments

The number of -1's among the four terms following the 0 at a(5*k), for k >= 0, is 1, 2, 3, 3, 1, 0, 3, 2, 2, 1, 4, 4, 0, 1, 3, 3, 1, 1, 3, 4, ...

See the Weisstein link, where it is stated that the period length is 0.

In general, the sequence {(k/n)} is not periodic if and only if k == 3 (mod 4). - Jianing Song, Dec 30 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..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 (contains this sequence).
• Wikipedia, Kronecker Symbol

Crossrefs

Cf. A035183 (inverse Moebius transform).

Sequences related to Kronecker symbols that do not form a Dirichlet character: this sequence {(-5/n)}, A034947 {(-1/n)}, A091338 {(3/n)}, A089509 {(7/n)}.

Cf. A080891 (5/n), A100047.

Programs

Maple

0, seq(numtheory:-jacobi(-5, n), n=1..89); # Peter Luschny, Dec 30 2018

Mathematica

Table[KroneckerSymbol[-5, n],{n,0,89}]

PARI

a(n)=kronecker(-5,n); \\ Andrew Howroyd, Jul 23 2018

Formula

Completely multiplicative with a(2) = -1, a(5) = 0, a(p) = 1 if p == 1, 3, 7, 9 (mod 20), a(p) = -1 if p == 11, 13, 17, 19 (mod 20). - Jianing Song, Dec 30 2018

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018