A construction asymptotic to Sqrt[8*n]*Log[Sqrt[8*n]]^2

(*start*)(*Mathematica*)Clear[a, f, p];
nn = 1000;
f[n_] := n;
a[n_] := DivisorSum[n, MoebiusMu[#] # &];
Monitor[TableForm[
   A = Accumulate[
     Table[Table[If[n >= k, a[GCD[n, k]], 0], {k, 1, nn}], {n, 1,
       nn}]]];, n]
TableForm[B = -Abs[A]];
Clear[A];
B[[All, 1]] = N[Table[f[n], {n, 1, nn}]];
TableForm[B];
TableForm[B1 = Sign[Transpose[Accumulate[Transpose[B]]]]];
Quiet[g4 =
  Show[ListLinePlot[
    v = ReplaceAll[
      Flatten[Table[First[Position[B1[[n]], -1]], {n, 1, nn}]],
      First[{}] -> 1]],
   Plot[Sqrt[8*f[n]], {n, 1, nn}, PlotStyle -> {Red, Thick}],
   ImageSize -> Large]]
Show[ListLinePlot[
  Table[Sum[
     1, {k, 1, v[[n]]}]*(Sum[1/k, {k, 1, v[[n]]}] - EulerGamma)^2, {n,
     1, nn}]],
 Plot[Sqrt[8*n]*Log[Sqrt[8*n]]^2, {n, 1, nn},
  PlotStyle -> {Red, Thick}], ImageSize -> Large]
(*end*)