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.

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, 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, 0, 0, 0, 0, 10, 0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32

Offset: 1

Author

Mats Granvik, Oct 08 2023

Keywords

• nonn
• tabl

Comments

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).

Examples

			{
{1}, = 1^2
{0, 4}, = 2^2
{5, 0, 4}, = 3^2
{0, 8, 0, 8}, = 4^2
{21, 0, 0, 0, 4}, = 5^2
{0, 20, 0, 0, 0, 16}, = 6^2
{40, 0, 0, 0, 0, 0, 9}, = 7^2
{0, 32, 0, 16, 0, 0, 0, 16}, = 8^2
{45, 0, 24, 0, 0, 0, 0, 0, 12}, = 9^2
{0, 84, 0, 0, 0, 0, 0, 0, 0, 16}, = 10^2
{111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10}, = 11^2
{0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32} = 12^2
}
		

Crossrefs

Cf. A127649, A366346, A000290, A060457, A002070, A036689, A001248, A272196.

Programs

Mathematica

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}]]

Formula

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.

Conjecture: T(n,n) = A060457(n).