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A134323 a(n) = Legendre(-3, prime(n)). 8
-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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

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

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

REFERENCES

Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, Vol 2, pp 173-175.

LINKS

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

Wikipedia, Balanced Ternary

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).

EXAMPLE

The twentieth prime is 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 of 71 is {1, 0, -1, 0, -1}, given that 1 * 3^4 + 0 * 3^3 - 1 * 3^2 + 0 * 3^1 - 1 * 3^0 = 71.

The twenty-first prime is 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 of 73 is {1, 0 -1, 0, 1}, as 1 * 3^4 + 0 * 3^3 - 1 * 3^2 + 0 * 3^1 + 1 * 3^0 = 73.

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-Fran├žois 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, A003627, A002476, A049084, A112632 (partial sums), A070750, A257834.

Sequence in context: A013596 A131695 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

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Last modified November 19 14:52 EST 2018. Contains 317352 sequences. (Running on oeis4.)