Asymptotics of intersection points of Harmonic numbers and reversed Harmonic numbers

(*start*)
(*Slow program*)
nnn = 300;
Monitor[a =
   Table[Sum[
     Sign[1 -
       Sign[HarmonicNumber[
          n] - (HarmonicNumber[nn] - HarmonicNumber[n - 1])]], {n, 1,
      nn}], {nn, 1, nnn}];, nn]
a[[Range[90]]]
g1 = ListLinePlot[a];
g2 = Plot[-(1/2) +
    1/2 E^(-EulerGamma/2) Sqrt[E^EulerGamma + 4 nn], {nn, 1,
    Length[a]}];
Show[g1, g2]
(*end*)

(*start*)
(*Faster program*)
nn = 1000;
Monitor[a =
   Table[Total[Sign[1 - Sign[Accumulate[Table[1/k, {k, 1, n}]] -
         Reverse[Accumulate[Reverse[Table[1/k, {k, 1, n}]]]]]]], {n,
     1, nn}];, n]
a[[Range[50]]]
g1 = ListLinePlot[a];
g2 = Plot[-(1/2) +
    1/2 E^(-EulerGamma/2) Sqrt[E^EulerGamma + 4 nn], {nn, 1,
    Length[a]}];
Show[g1, g2]
(*end*)