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.
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, 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, 0, 0, 0, 0, 0, 21, 32, 0, 40, 32, 0, 0, 0, 0, 0, 0, 0, 40
Offset: 1
Examples
{
{1}, = 1^2
{2, 2}, = 2^2
{4, 0, 5}, = 3^2
{8, 0, 0, 8}, = 4^2
{16, 0, 0, 0, 9}, = 5^2
{8, 8, 10, 0, 0, 10}, = 6^2
{36, 0, 0, 0, 0, 0, 13}, = 7^2
{32, 0, 0, 8, 0, 0, 0, 24}, = 8^2
{36, 0, 24, 0, 0, 0, 0, 0, 21}, = 9^2
{32, 32, 0, 0, 18, 0, 0, 0, 0, 18}, = 10^2
{100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21}, = 11^2
{32, 0, 40, 32, 0, 0, 0, 0, 0, 0, 0, 40} = 12^2
}
Programs
-
Mathematica
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}]] -
PARI
T(n,k) = sum(x=1, n, sum(y=1, n, gcd(x^2 - y^2, n) == k)); \\ Michel Marcus, Oct 14 2023
Formula
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.
Conjecture 1: T(n,n) = A062803(n).
Conjecture 2: T(n,1) = A082953(n).
Conjectures from Ridouane Oudra, Jun 17 2025: (Start)
T(n,k) = 0 iff k not divide n.
T(n,k) = phi(n/k)*Sum_{d|k} (k/d)*phi(d*n/k), for n odd and k|n.
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.
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.
Comments