A381884 Triangle read by rows: T(n, k) = 0 if n = 0 or k is not a quadratic residue modulo n, otherwise T(n, k) = k.

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

Offset

0,6

Links

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

Example

Triangle T(n, k) starts:
  [0] 0;
  [1] 0, 1;
  [2] 0, 1, 2;
  [3] 0, 1, 0, 3;
  [4] 0, 1, 0, 0, 4;
  [5] 0, 1, 0, 0, 4, 5;
  [6] 0, 1, 0, 3, 4, 0, 6;
  [7] 0, 1, 2, 0, 4, 0, 0, 7;
  [8] 0, 1, 0, 0, 4, 0, 0, 0, 8;
  [9] 0, 1, 0, 0, 4, 0, 0, 7, 0, 9;
.
Array Arow(n) = [T(j, n), j = 0.. ] starts:
  [0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  [1] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  [2] 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, ...
  [3] 0, 3, 3, 3, 0, 0, 3, 0, 0, 0, ...
  [4] 0, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
  [5] 0, 5, 5, 0, 5, 5, 0, 0, 0, 0, ...
  [6] 0, 6, 6, 6, 0, 6, 6, 0, 0, 0, ...
  [7] 0, 7, 7, 7, 0, 0, 7, 7, 0, 7, ...
  [8] 0, 8, 8, 0, 8, 0, 0, 8, 8, 0, ...
  [9] 0, 9, 9, 9, 9, 9, 9, 9, 9, 9, ...
.
3 is not a quadratic residue modulo 7, therefore T(7, 3) = 0.
12 is a quadratic residue modulo 23, therefore T(23, 12) = 12.

Maple

QR := (k, n) -> ifelse(n = 0 or NumberTheory:-QuadraticResidue(k, n) < 0, 0, 1):
T := (n, k) -> k*QR(k, n): seq(seq(T(n, k), k = 0..n), n = 0..12);
Arow := (n, len) -> local j; seq(T(j, n), j=0..len):
seq(lprint([n], Arow(n, 9)), n=0..9);

Mathematica

QR[n_, k_] := Module[{x, y}, If[Reduce[x^2 == n + k*y, {x, y}, Integers] =!= False, 1, -1]];
T[n_, k_] := If[n == 0 || QR[k, n] < 0, 0, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

Prog

(Python)
from sympy.ntheory import is_quad_residue
def QR(n, k): return is_quad_residue(n, k)
def T(n, k): return 0 if n == 0 or not QR(k, n) else k
for n in range(13): print([T(n, k) for k in range(n + 1)])

Crossrefs

Indices of the nonzero terms in row n of the array: A057126 (row 2), A057125 (row 3), A057762 (row 5), A262931 (row 6), A262932 (row 7).

Sequence in context: A356945 A162590 A393311 * A276424 A346070 A191258

Adjacent sequences: A381881 A381882 A381883 * A381885 A381886 A381887

Keyword

nonn,tabl

Author

Peter Luschny, Mar 17 2025

Status

approved