A260238 Denominators of the characteristic polynomials of the von Mangoldt function matrix.

1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 2, 4, 12, 1, 1, 2, 20, 3, 60, 1, 1, 2, 60, 360, 360, 60, 1, 1, 2, 420, 2520, 1260, 504, 420, 1, 1, 1, 4, 120, 5040, 2520, 5040, 840, 1, 1, 1, 1, 3, 360, 15120, 7560, 15120, 2520, 1

Offset: 1

Author

Mats Granvik, Jul 20 2015

Keywords

• tabl
• nonn
• frac

Comments

The von Mangoldt function matrix is the symmetric Greatest Common Divisor (GCD) type matrix A191898 divided by either the row index or the column index.

Every eigenvalue of a smaller von Mangoldt function matrix appears to be common to infinitely many larger von Mangoldt matrices. The eigenvalues of smaller von Mangoldt function matrices also repeat within larger von Mangoldt function matrices.

The second diagonal appears to be A003418.

Examples

			{
{0},
{1, 1},
{1, 2, 1},
{1, 6, 6, 1},
{1, 2, 4, 12, 1},
{1, 2, 20, 3, 60, 1},
{1, 2, 60, 360, 360, 60, 1},
{1, 2, 420, 2520, 1260, 504, 420, 1},
{1, 1, 4, 120, 5040, 2520, 5040, 840, 1},
{1, 1, 1, 3, 360, 15120, 7560, 15120, 2520, 1}
}
		

Crossrefs

Cf. A191898, A003418. Numerators in A260237.

Programs

Mathematica

Clear[nnn, nn, T, n, k, x]; nnn = 9; T[n_, k_] := T[n, k] = Which[n < 1 || k < 1, 0, n == 1 || k == 1, 1, k > n, T[k, n], n > k, T[k, Mod[n, k, 1]], True, -Sum[T[n, i], {i, n - 1}]];b = Table[CoefficientList[CharacteristicPolynomial[Table[Table[T[n, k]/n, {k, 1, nn}], {n, 1, nn}], x], x], {nn, 1, nnn}];Flatten[Denominator[b]]