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I am having a little trouble calculating the Dickman Function from scratch. I have downloaded the notebook from the above link and the have following code:

Off[
 InterpolatingFunction::dmval, NDSolve::precw, Power::"infy", Less::"nord", NDSolve::"nlnum"
 ];

Clear[F];
F[0][x_] = Which[x < 0, 0, 0 <= x <= 1, Exp[1 - 1/x], x > 1, 1];
With[{ε = 10^-5, n = 20},
  Do[ndsol[k] = 
    NDSolve[{F[k]'[x] == F[k - 1][x/(1 - x)]/x, F[k][ε] == F[k - 1][ε]}, 
      F[k], {x, ε, 1},
      WorkingPrecision -> 35, MaxSteps -> 10^5, PrecisionGoal -> 18, 
      AccuracyGoal -> 18][[1, 1, 2]];
    Print["Step ", k, "..."];
    F[k][x_] = 
      Which[
        x < 0, 0, 
        0 <= x <= 1, Evaluate[ndsol[k][x]/ndsol[k][2]], 
        x > 1, 1],
    {k, n}]]
Plot[F[20][a], {a, 0, 1}, PlotRange -> All, PlotStyle -> Red]

It makes the following plot:

enter image description here

which is a plot of $$\rm {F}(\alpha)=\int_{0}^{\alpha}\rm {F} \bigg ( \frac{t}{1-t} \bigg)\frac{\,dt}{t}$$

I am having trouble modifying the above Mathematica code so I can eventually plot

$$\rm {G}(\beta)=\int_{0}^{\beta}\bigg [\rm {G} \bigg (\frac{t}{1-t} \bigg )-\rm {F} \bigg (\frac{t}{1-t} \bigg ) \bigg]\frac{\,dt}{t}$$

using the same method. Ultimately, I would like to go on to plot

$$\rm {H}(\gamma)=\int_{0}^{\gamma}\bigg [\rm {H} \bigg (\frac{t}{1-t} \bigg )-\rm {G} \bigg (\frac{t}{1-t} \bigg )-\rm {F} \bigg (\frac{t}{1-t} \bigg ) \bigg]\frac{\,dt}{t}$$

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