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

1,8

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.

Links

Table of n, a(n) for n=1..55.

Example

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}
}

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]}]

Crossrefs

Denominators in A260238.

Sequence in context: A294254 A216792 A359215 * A118135 A318671 A337335

Adjacent sequences: A260234 A260235 A260236 * A260238 A260239 A260240

Keyword

tabl,sign,frac

Author

Mats Granvik, Jul 20 2015

Status

approved