A134323 a(n) = Legendre(-3, prime(n)).

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

Offset

1,1

Comments

Value of lowest trit of prime(n) in balanced ternary representation (A059095) (original definition).

For p = prime(n) != 3, a(n) = +1 if p is of the form 3*k + 1, and -1 if the p is of the form 3*k - 1. - Joerg Arndt, Sep 16 2014

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

-1 if the n-th prime is 2 or == 5 mod 6, +1 if the n-th prime is == 1 mod 6, and 0 if it is 3.

a(n) = (1 - 0^A039701(n)) * (-1)^(A039701(n)+1).

a(n) != 0 for n != 2;

a(A049084(A003627(n))) = -1; a(A049084(A002476(n))) = +1.

Example

For n=20, prime(20) = 71, and we verify that -3 is not a quadratic residue modulo 71, hence a(20) = -1. Also, we see that the balanced ternary representation row A059095(71) = {1, 0, -1, 0, -1} which ends in -1.
For n=21, prime(21) = 73, and we see that x^2 = -3 mod 73 has solutions like x = 17, 56, hence a(21) = 1. Also, the balanced ternary representation row A059095(73) = {1, 0 -1, 0, 1} which ends in 1.

Mathematica

A134323[n_] := (r = Mod[Prime[n], 6]; If[r == 1, 1, -1]); A134323[1] = -1; A134323[2] = 0; Table[A134323[n], {n, 1, 102}] (* Jean-François Alcover, Nov 07 2011, after Bill McEachen *)
JacobiSymbol[-3, Prime[Range[100]]] (* Alonso del Arte, Aug 02 2017 *)

Prog

(Haskell)
a134323 n = (1 - 0 ^ m) * (-1) ^ (m + 1) where m = a000040 n `mod` 3
-- Reinhard Zumkeller, Sep 16 2014
(PARI) apply(p->kronecker(-3, p), primes(100)) \\ Charles R Greathouse IV, Aug 14 2017

Crossrefs

Cf. A000040, A039701, A049084, A112632 (partial sums), A059095 (balanced ternary)

Cf. A091177 (indices of -1's), A091178 (indices of +1's), A003627, A002476.

Other moduli: A070750, A257834.

Sequence in context: A131695 A324113 A105812 * A060576 A261012 A019590

Adjacent sequences: A134320 A134321 A134322 * A134324 A134325 A134326

Keyword

sign,easy

Author

Reinhard Zumkeller, Oct 21 2007

Extensions

Wording of definition changed by N. J. A. Sloane, Jun 21 2015

Name simplified by Alonso del Arte, Aug 02 2017

Status

approved