Möbius function approximation by Matrix exponential

Clear[A, a, n, k, nn];
nn = 600;
a[n_] := Total[Divisors[n]*MoebiusMu[Divisors[n]]];
Monitor[A =
   Table[If[n == 1, 0, Sum[a[GCD[n, k]]/N[k]/Log[n], {k, 1, nn}]], {n,
      1, nn}];, n]
ListPlot[b =
   MatrixExp[
     Table[Table[If[Mod[n, k] == 0, -A[[n/k]], 0], {k, 1, nn}], {n, 1,
        nn}]][[All, 1]], Filling -> 0];
ListLinePlot[Accumulate[b]]
ListPlot[b, Filling -> Axis]
Round[b[[Range[42]]]]