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
Examples
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.
Crossrefs
Programs
-
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) -> 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 -
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)])