(*Mathematica start*) (*Program to compute the sequence with the conjectured asymptotic sqrt(8*n)*) Clear[a, b, f, p]; nn = 2000; f[n_] := n; a[n_] := DivisorSum[n, MoebiusMu[#] # &]; Monitor[TableForm[ A = Accumulate[ Table[Table[a[GCD[n, k]], {k, 1, nn}], {n, 1, nn}]]];, n] TableForm[B = -Abs[A]]; Clear[A]; B[[All, 1]] = Table[f[n], {n, 1, nn}]; TableForm[B]; b = 1 + Total[Sign[1 + Sign[Accumulate[Transpose[B]]]]]; Clear[B]; Show[ListLinePlot[b], Plot[Sqrt[8*n], {n, 1, nn}, PlotStyle -> {Red, Thick}]] (*Mathematica end*)