A366444 Triangle read by rows: T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

1, 1, -1, 2, 0, -2, 2, -1, 0, -1, 4, 0, 0, 0, -4, 2, -2, -2, 0, 0, 2, 6, 0, 0, 0, 0, 0, -6, 4, -2, 0, -1, 0, 0, 0, -1, 6, 0, -4, 0, 0, 0, 0, 0, -2, 4, -4, 0, 0, -4, 0, 0, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 4, -2, -4, -2, 0, 2, 0, 0, 0, 0, 0, 2

Offset: 1

Author

Mats Granvik, Oct 12 2023

Keywords

• sign
• tabl

Comments

Sum_{k=1..n} T(n,k) = A063524(n).

Examples

			{
{1}, = 1
{1, -1}, = 0
{2, 0, -2}, = 0
{2, -1, 0, -1}, = 0
{4, 0, 0, 0, -4}, = 0
{2, -2, -2, 0, 0, 2}, = 0
{6, 0, 0, 0, 0, 0, -6}, = 0
{4, -2, 0, -1, 0, 0, 0, -1}, = 0
{6, 0, -4, 0, 0, 0, 0, 0, -2}, = 0
{4, -4, 0, 0, -4, 0, 0, 0, 0, 4}, = 0
{10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10}, = 0
{4, -2, -4, -2, 0, 2, 0, 0, 0, 0, 0, 2} = 0
}
		

Crossrefs

Cf. A000010, A023900, A366445, A054524, A054525, A063524, A054522, A054523, A129691, A127649.

Programs

Mathematica

nn = 12; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[If[Mod[n, k] == 0, EulerPhi[n/k]*g[k], 0], {k, 1, n}], {n, 1, nn}]]

Formula

T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).