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A366362 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

%I A366362 #19 Oct 08 2023 09:03:25
%S A366362 1,0,4,5,0,4,0,8,0,8,21,0,0,0,4,0,20,0,0,0,16,40,0,0,0,0,0,9,0,32,0,
%T A366362 16,0,0,0,16,45,0,24,0,0,0,0,0,12,0,84,0,0,0,0,0,0,0,16,111,0,0,0,0,0,
%U A366362 0,0,0,0,10,0,40,0,40,0,32,0,0,0,0,0,32
%N A366362 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.
%C A366362 Row n appears to have sum n^2. T(prime(m),1) = A366346(m). The number of nonzero terms in row n appears to be A320111(n).
%F A366362 T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.
%F A366362 Conjecture: T(n,n) = A060457(n).
%e A366362 {
%e A366362 {1}, = 1^2
%e A366362 {0, 4}, = 2^2
%e A366362 {5, 0, 4}, = 3^2
%e A366362 {0, 8, 0, 8}, = 4^2
%e A366362 {21, 0, 0, 0, 4}, = 5^2
%e A366362 {0, 20, 0, 0, 0, 16}, = 6^2
%e A366362 {40, 0, 0, 0, 0, 0, 9}, = 7^2
%e A366362 {0, 32, 0, 16, 0, 0, 0, 16}, = 8^2
%e A366362 {45, 0, 24, 0, 0, 0, 0, 0, 12}, = 9^2
%e A366362 {0, 84, 0, 0, 0, 0, 0, 0, 0, 16}, = 10^2
%e A366362 {111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10}, = 11^2
%e A366362 {0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32} = 12^2
%e A366362 }
%t A366362 f = x^3 - x^2 - y^2 - y; nn = 12; Flatten[Table[Table[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]]
%Y A366362 Cf. A127649, A366346, A000290, A060457, A002070, A036689, A001248, A272196.
%K A366362 nonn,tabl
%O A366362 1,3
%A A366362 _Mats Granvik_, Oct 08 2023