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A268921 Irregular triangle with the Legendre symbol (-m / prime(n)) for m = 0,1, ..., prime(n)-1, for n >= 1. Caution for row n = 1. 0
0, 1, 0, -1, 1, 0, 1, -1, -1, 1, 0, -1, -1, 1, -1, 1, 1, 0, -1, 1, -1, -1, -1, 1, 1, 1, -1, 1, 0, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, 0, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 0, -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

COMMENTS

Row n has length prime(n) = A000040(n).

If GCD(-m, prime(n)) is not 1 then the Legendre and Jacobi symbols are put to 0. Therefore T(0, prime(n)) = 0, for n >= 1.

Because for GCD(-a,prime(n)) = 1  and  for n >= 2 the Legendre symbol is (-a)^((prime(n)-1)/2) (mod prime(n)), it is sufficient to consider a = 0 .. prime(n) - 1, due to periodicity.

Caution for n=1 (prime 2): Jacobi(-a/2) has period length 8: [0,1,0,-1,0,-1,0,1]. Here row n = 1 is [0, 1]. For odd -m the solution of x^2 == -1 (mod 2) is x = 1 in the residue class {0,1} modulo 2. So -m is always a quadratic residue modulo 2 for odd m. This would lead to [repeat (0,1)] with period length 2.

LINKS

Table of n, a(n) for n=1..77.

FORMULA

T(n, m) = 0 if m = 0. T(n, m) = Legendre(-m, prime(n)) for m = 1, ..., prime(n)-1, n >= 2, and T(1, 1) = +1 (Jacobi symbol).

EXAMPLE

The irregular triangle T(n, m) begins (here P(n) = prime(n)):

n, P(n)\m 0  1  2  3  4  5  6  7  8  9 10 ...

1,  2:    0  1

2,  3:    0 -1  1

3,  5:    0  1 -1 -1  1

4,  7:    0 -1 -1  1 -1  1  1

5, 11:    0 -1  1 -1 -1 -1  1  1  1 -1  1

...

Row n=6, P(6)=13: 0  1  1 -1  1 -1 -1 -1  1  1 -1 -1 -1 1 -1 1 1;

Row n=7, P(7)=17: 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1;

Row n=8, P(8)=19: 0 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 1.

...

CROSSREFS

Cf. A000040, A226520.

Sequence in context: A157412 A023532 A226520 * A327180 A030308 A280237

Adjacent sequences:  A268918 A268919 A268920 * A268922 A268923 A268924

KEYWORD

sign,tabf

AUTHOR

Wolfdieter Lang, Feb 29 2016

STATUS

approved

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Last modified May 30 04:38 EDT 2020. Contains 334711 sequences. (Running on oeis4.)