A368197 Triangle read by rows: T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k], where f(x,y,z) = x^2 + y^2 - z^2.

1, 4, 4, 18, 0, 9, 32, 8, 0, 24, 100, 0, 0, 0, 25, 72, 72, 36, 0, 0, 36, 294, 0, 0, 0, 0, 0, 49, 256, 64, 0, 96, 0, 0, 0, 96, 486, 0, 144, 0, 0, 0, 0, 0, 99, 400, 400, 0, 0, 100, 0, 0, 0, 0, 100, 1210, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121

Offset: 1

Author

Mats Granvik, Dec 16 2023

Keywords

• nonn
• tabl

Comments

Row n has sum n^3. The number of nonzero terms in row n appears to be A000005(n). It appears that Sum_{k=1..n} T(n,k)*A023900(k) = A063524(n). Main diagonal appears to be A062775. First column appears to be A053191.

It appears that when p > 2 in f(x,y,z,p) = x^p + y^p - z^p and T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z,p), n) = k], then Sum_{k=1..n} T(n,k)*A023900(k) is not equal to A063524(n). - Mats Granvik, May 07 2024

Examples

			Triangle begins:
     1;
     4,   4;
    18,   0,   9;
    32,   8,   0,  24;
   100,   0,   0,   0,  25;
    72,  72,  36,   0,   0,  36;
   294,   0,   0,   0,   0,   0,  49;
   256,  64,   0,  96,   0,   0,   0,  96;
   486,   0, 144,   0,   0,   0,   0,   0,  99;
   400, 400,   0,   0, 100,   0,   0,   0,   0, 100;
  1210,   0,   0,   0,   0,   0,   0,   0,   0,   0, 121;
  ...
		

Crossrefs

Cf. A000578, A000005, A023900, A063524, A062775, A053191, A367689.

Programs

Mathematica

nn = 11; p = 2; f = x^p + y^p - z^p; Flatten[Table[Table[Sum[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {z, 1, n}], {k, 1, n}], {n, 1, nn}]]

Formula

T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k], where f(x,y,z) = x^2 + y^2 - z^2.