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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.

%I A368197 #39 Aug 25 2024 18:57:24
%S A368197 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,
%T A368197 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,
%U A368197 100,1210,0,0,0,0,0,0,0,0,0,121
%N 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.
%C A368197 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.
%C A368197 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
%F A368197 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.
%e A368197 Triangle begins:
%e A368197      1;
%e A368197      4,   4;
%e A368197     18,   0,   9;
%e A368197     32,   8,   0,  24;
%e A368197    100,   0,   0,   0,  25;
%e A368197     72,  72,  36,   0,   0,  36;
%e A368197    294,   0,   0,   0,   0,   0,  49;
%e A368197    256,  64,   0,  96,   0,   0,   0,  96;
%e A368197    486,   0, 144,   0,   0,   0,   0,   0,  99;
%e A368197    400, 400,   0,   0, 100,   0,   0,   0,   0, 100;
%e A368197   1210,   0,   0,   0,   0,   0,   0,   0,   0,   0, 121;
%e A368197   ...
%t A368197 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}]]
%Y A368197 Cf. A000578, A000005, A023900, A063524, A062775, A053191, A367689.
%K A368197 nonn,tabl
%O A368197 1,2
%A A368197 _Mats Granvik_, Dec 16 2023