A373748 Triangle read by rows: T(n, k) is k if k is a quadratic residue modulo n, otherwise is -k and is a quadratic nonresidue modulo n. T(0, 0) = 0 by convention.

0, 0, 1, 0, 1, 2, 0, 1, -2, 3, 0, 1, -2, -3, 4, 0, 1, -2, -3, 4, 5, 0, 1, -2, 3, 4, -5, 6, 0, 1, 2, -3, 4, -5, -6, 7, 0, 1, -2, -3, 4, -5, -6, -7, 8, 0, 1, -2, -3, 4, -5, -6, 7, -8, 9, 0, 1, -2, -3, 4, 5, 6, -7, -8, 9, 10, 0, 1, -2, 3, 4, 5, -6, -7, -8, 9, -10, 11, 0, 1, -2, -3, 4, -5, -6, -7, -8, 9, -10, -11, 12

Offset

0,6

Links

Table of n, a(n) for n=0..90.

Carl Friedrich Gauss, Vierter Abschnitt. Von den Congruenzen zweiten Grades. Quadratische Reste und Nichtreste. Art. 97, in "Untersuchungen über die höhere Arithmetik", Hrsg. H. Maser, Verlag von Julius Springer, Berlin, 1889.

Peter Luschny, SageMath: is_quadratic_residue.

Example

Triangle starts:
  [0] [0]
  [1] [0,  1]
  [2] [0,  1,  2]
  [3] [0,  1, -2,  3]
  [4] [0,  1, -2, -3,  4]
  [5] [0,  1, -2, -3,  4,  5]
  [6] [0,  1, -2,  3,  4, -5,  6]
  [7] [0,  1,  2, -3,  4, -5, -6,  7]
  [8] [0,  1, -2, -3,  4, -5, -6, -7,  8]
  [9] [0,  1, -2, -3,  4, -5, -6,  7, -8,  9]
 [10] [0,  1, -2, -3,  4,  5,  6, -7, -8,  9,  10]

Maple

QR := (a, n) -> ifelse(n = 0, 1, NumberTheory:-QuadraticResidue(a, n)):
for n from 0 to 10 do seq(a*QR(a, n), a = 0..n) od;

Mathematica

qr[n_] := qr[n] = Join[Table[PowerMod[k, 2, n], {k, 0, Floor[n/2]}], {n}];
T[0, 0] := 0; T[n_, k_] := If[MemberQ[qr[n], k], k, -k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten

Prog

(SageMath)
def Trow(n):
    q = set(mod(a * a, n) for a in range(n // 2  + 1)).union({n})
    return [k if k in q else -k for k in range(n + 1)]
for n in range(11): print(Trow(n))

Crossrefs

Signed version of A002262.

Cf. A000004 (column 0), A001477 (main diagonal), A255644(n) + n (row sums).

Cf. A096008, A096013, A373749.

Sequence in context: A166352 A025690 A025668 * A048151 A025675 A025682

Adjacent sequences: A373745 A373746 A373747 * A373749 A373750 A373751

Keyword

sign,tabl

Author

Peter Luschny, Jun 27 2024

Status

approved