1,4

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.

{

{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}

}

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]]

Cf. A191898, A003418. Numerators in A260237.

Sequence in context: A166919 A338874 A338876 * A283795 A168641 A255914

Adjacent sequences: A260235 A260236 A260237 * A260239 A260240 A260241

tabl,nonn,frac

Mats Granvik, Jul 20 2015

approved