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A366561 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^2 - y^2.

%I A366561 #14 Jun 23 2025 22:59:18
%S A366561 1,2,2,4,0,5,8,0,0,8,16,0,0,0,9,8,8,10,0,0,10,36,0,0,0,0,0,13,32,0,0,
%T A366561 8,0,0,0,24,36,0,24,0,0,0,0,0,21,32,32,0,0,18,0,0,0,0,18,100,0,0,0,0,
%U A366561 0,0,0,0,0,21,32,0,40,32,0,0,0,0,0,0,0,40
%N A366561 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^2 - y^2.
%C A366561 Row n appears to have sum n^2. The number of nonzero terms in row n is A366563(n). Sum_{k=1..n} T(n,k)*A023900(k)/n = A366562(n).
%F A366561 T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^2 - y^2.
%F A366561 Conjecture 1: T(n,n) = A062803(n).
%F A366561 Conjecture 2: T(n,1) = A082953(n).
%F A366561 Conjectures from _Ridouane Oudra_, Jun 17 2025: (Start)
%F A366561 T(n,k) = 0 iff k not divide n.
%F A366561 T(n,k) = phi(n/k)*Sum_{d|k} (k/d)*phi(d*n/k), for n odd and k|n.
%F A366561 T(n,k) = 2*(-1)^k*phi(n/k)*Sum_{d|k} (-1)^(k/d)*(k/d)*phi(d*n/k), for n even and k|n.
%F A366561 T(n,k) = gcd(n,2)*(-1)^k*phi(n/k)*Sum_{d|k} (-1)^(k/d)*(k/d)*phi(d*n/k), for all integers n and k|n.
%F A366561 More generally, for all integers n, k: T(n,k) = gcd(n,2)*(-1)^k*A054523(n,k)*Sum_{d|k} (-1)^(k/d)*(k/d)*A054523(d*n,k). (End)
%e A366561 {
%e A366561 {1}, = 1^2
%e A366561 {2, 2}, = 2^2
%e A366561 {4, 0, 5}, = 3^2
%e A366561 {8, 0, 0, 8}, = 4^2
%e A366561 {16, 0, 0, 0, 9}, = 5^2
%e A366561 {8, 8, 10, 0, 0, 10}, = 6^2
%e A366561 {36, 0, 0, 0, 0, 0, 13}, = 7^2
%e A366561 {32, 0, 0, 8, 0, 0, 0, 24}, = 8^2
%e A366561 {36, 0, 24, 0, 0, 0, 0, 0, 21}, = 9^2
%e A366561 {32, 32, 0, 0, 18, 0, 0, 0, 0, 18}, = 10^2
%e A366561 {100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21}, = 11^2
%e A366561 {32, 0, 40, 32, 0, 0, 0, 0, 0, 0, 0, 40} = 12^2
%e A366561 }
%t A366561 nn = 12; f = x^2 - y^2; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]]
%o A366561 (PARI) T(n,k) = sum(x=1, n, sum(y=1, n, gcd(x^2 - y^2, n) == k)); \\ _Michel Marcus_, Oct 14 2023
%Y A366561 Cf. A127649, A000290, A062803, A082953.
%Y A366561 Cf. A000010, A054523.
%K A366561 nonn,tabl
%O A366561 1,2
%A A366561 _Mats Granvik_, Oct 13 2023