A372726 Legendre's triangle read by rows. T(n, k) = L(n / prime(k)) where L(n/p) is the Legendre symbol, for n >= 0 and 2 <= k <= n + 2.

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

Offset

0

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).

Adrien Marie Legendre, Essai sur la théorie des nombres, Paris, Duprat, an VI [1798]. Introducing the symbol, p. 186.

Formula

T(n, k) = r - p*[r > 1] where r = n^v mod p, p = prime(k), v = (p - 1)/2, and [.] are the Iverson brackets.

Example

Triangle starts:
  [ 0]  0;
  [ 1]  1,  1;
  [ 2] -1, -1,  1;
  [ 3]  0, -1, -1,  1;
  [ 4]  1,  1,  1,  1,  1;
  [ 5] -1,  0, -1,  1, -1, -1;
  [ 6]  0,  1, -1, -1, -1, -1,  1;
  [ 7]  1, -1,  0, -1, -1, -1,  1, -1;
  [ 8] -1, -1,  1, -1, -1,  1, -1,  1, -1;
  [ 9]  0,  1,  1,  1,  1,  1,  1,  1,  1,  1;
  [10]  1,  0, -1, -1,  1, -1, -1, -1, -1,  1,  1;
.
Not limiting the range of k leads to the square array:
.
  [n\p] 3,  5,  7, 11, 13, 17, 19, 23, 29, 31
  -----------------------------------------------
  [0]   0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
  [1]   1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  [2]  -1, -1,  1, -1, -1,  1, -1,  1, -1,  1, ...
  [3]   0, -1, -1,  1,  1, -1, -1,  1, -1, -1, ...
  [4]   1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  [5]  -1,  0, -1,  1, -1, -1,  1, -1,  1,  1, ...
  [6]   0,  1, -1, -1, -1, -1,  1,  1,  1, -1, ...
  [7]   1, -1,  0, -1, -1, -1,  1, -1,  1,  1, ...
  [8]  -1, -1,  1, -1, -1,  1, -1,  1, -1,  1, ...
  [9]   0,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  ...

Maple

L := (n, k) -> NumberTheory:-LegendreSymbol(n, ithprime(k)):
for n from 0 to 10 do lprint([n], seq(L(n, k), k = 2..n + 2)) od;

Mathematica

Array[JacobiSymbol[#, Prime[Range[2, #+2]]]&, 15, 0] (* Paolo Xausa, Jul 09 2024 *)

Prog

(Python)
from sympy import primerange, prime, legendre_symbol
for n in range(11):
    print([n], [legendre_symbol(n, p) for p in primerange(3, prime(n + 2) + 1)])
# For illustration of the formula (Sympy's implementation is more efficent):
def LegendreSymbol(n, p):
    v = (p - 1) // 2
    res = pow(n, v, p)
    return res - p if res > 1 else res

Crossrefs

Family: A217831 (Euclid's triangle), A372877 (Jacobi's triangle), A372728 (Kronecker's triangle), A373223 (Gauss' triangle), A373751 (quadratic residue modulo prime(n)), A373748 (quadratic residue/nonresidue modulo n).

Cf. A372725 (row sums).

Sequence in context: A182067 A196147 A097325 * A242647 A167393 A275606

Adjacent sequences: A372723 A372724 A372725 * A372727 A372728 A372729

Keyword

sign,tabl

Author

Peter Luschny, May 22 2024

Extensions

Data corrected by Paolo Xausa, Jul 09 2024

Status

approved