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

Author

Reinhard Zumkeller, Oct 21 2007

Keywords

• sign
• easy

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

Examples

			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.
		

Links

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

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.

Cf. A102283.

Programs

Haskell

a134323 n = (1 - 0 ^ m) * (-1) ^ (m + 1) where m = a000040 n `mod` 3
-- Reinhard Zumkeller, Sep 16 2014

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 *)

PARI

apply(p->kronecker(-3,p), primes(100)) \\ Charles R Greathouse IV, Aug 14 2017

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.

a(n) = A102283(prime(n)). - Ridouane Oudra, Jan 09 2025

Extensions

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

Name simplified by Alonso del Arte, Aug 02 2017