A260237 Numerators of the characteristic polynomials of the von Mangoldt function matrix.

0, 1, -1, -1, -1, 1, 1, 11, -1, -1, 0, -3, -9, 5, 1, 0, 3, 81, 7, -73, -1, 0, 3, 73, -1261, -1183, 53, 1, 0, -3, -1231, 5251, 8989, 1451, -731, -1, 0, 0, 7, 397, -12491, -19877, -15047, 1567, 1, 0, 0, 0, -7, -1483, 50111, 69761, 45959, -5261, -1

Offset: 1

Author

Mats Granvik, Jul 20 2015

Keywords

• tabl
• sign
• frac

Comments

The von Mangoldt function matrix is the symmetric 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.

Examples

			The first term set to zero is not part of the characteristic polynomials. It is only there for the formatting of the table.
{
{0},
{1, -1},
{-1, -1, 1},
{1, 11, -1, -1},
{0, -3, -9, 5, 1},
{0, 3, 81, 7, -73, -1},
{0, 3, 73, -1261, -1183, 53, 1},
{0, -3, -1231, 5251, 8989, 1451, -731, -1},
{0, 0, 7, 397, -12491, -19877, -15047, 1567, 1},
{0, 0, 0, -7, -1483, 50111, 69761, 45959, -5261,-1}
}
		

Crossrefs

Denominators in A260238.

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[{0,Numerator[b]}]