[seqfan] A recurrence for the Dirichlet inverse of the Euler totient function.

[Mats Granvik]

This Mathematica program generates the Dirichlet inverse of the Euler  
totient function as the main diagonal of a symmetric array with  
periodic columns and rows: https://oeis.org/A023900. The recurrence is  
related to the recurrence for the Mahonian numbers:  
https://oeis.org/A008302

By changing signs in front of the sums we can get the
Fredholm-Rueppel sequence. https://oeis.org/A036987

Mats Granvik

Mathematica program:

Clear[t, n, k, a, b];
nn = 50;
t[n_, 1] = 1;
t[1, k_] = 1;
t[n_, k_] :=
   t[n, k] =
    If[n < k,
     If[And[n > 1, k > 1], Sum[-t[k - i, n], {i, 1, n - 1}], 0],
     If[And[n > 1, k > 1], Sum[-t[n - i, k], {i, 1, k - 1}], 0]];
a = Flatten[Table[Table[t[n, k], {k, n, n}], {n, 1, nn}]];
a
Sign[a]
b = MoebiusMu[
    Prepend[Array[Times @@ First[Transpose[FactorInteger[#]]] &,
      nn - 1, 2], 1]];
Sign[a] - b