A366561 - OEIS

%I #10 Oct 14 2023 14:07:00

%S 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 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 0,0,0,0,0,21,32,0,40,32,0,0,0,0,0,0,0,40

%N 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 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 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 Conjecture 1: T(n,n) = A062803(n).

%F Conjecture 2: T(n,1) = A082953(n).

%e {

%e {1}, = 1^2

%e {2, 2}, = 2^2

%e {4, 0, 5}, = 3^2

%e {8, 0, 0, 8}, = 4^2

%e {16, 0, 0, 0, 9}, = 5^2

%e {8, 8, 10, 0, 0, 10}, = 6^2

%e {36, 0, 0, 0, 0, 0, 13}, = 7^2

%e {32, 0, 0, 8, 0, 0, 0, 24}, = 8^2

%e {36, 0, 24, 0, 0, 0, 0, 0, 21}, = 9^2

%e {32, 32, 0, 0, 18, 0, 0, 0, 0, 18}, = 10^2

%e {100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21}, = 11^2

%e {32, 0, 40, 32, 0, 0, 0, 0, 0, 0, 0, 40} = 12^2

%e }

%t 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 (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 Cf. A127649, A000290, A062803, A082953.

%K nonn,tabl

%O 1,2

%A _Mats Granvik_, Oct 13 2023