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Setting:

$$F(x)=x\log(x)+\sum _{j=2}^k -|A(x,j)| \tag{1}$$

appears to give the asymptotic $\sqrt{8x\log(x)}$ for the least $k$ such that $F(x)$ is negative.

In general it appears that the least $k$ such that:

$$F(x)=f(x)+\sum _{j=2}^k -|A(x,j)| \tag{2}$$

is negative, has the asymptotic: $\sqrt{8f(x)}$.

Mathematica program at pastebin: https://pastebin.com/GJ81MQez

Setting:

$$F(x)=x\log(x)+\sum _{j=2}^k -|A(x,j)| \tag{1}$$

appears to give the asymptotic $\sqrt{8x\log(x)}$ for the least $k$ such that $F(x)$ is negative.

In general it appears that the least $k$ such that:

$$F(x)=f(x)+\sum _{j=2}^k -|A(x,j)| \tag{2}$$

is negative, has the asymptotic: $\sqrt{8f(x)}$.

See the Mathematica program in the question, by setting p=1.

Setting:

$$F(x)=\frac{x\log(x)}{8}+\sum _{j=2}^k -|A(x,j)| \tag{1}$$

appears to give the asymptotic $\sqrt{x\log(x)}$ for the least $k$ such that $F(x)$ is negative.

In general it appears that the least $k$ such that:

$$F(x)=\frac{f(x)}{8}+\sum _{j=2}^k -|A(x,j)| \tag{2}$$

is negative, has the asymptotic: $\sqrt{f(x)}$.

Mathematica program at pastebin: https://pastebin.com/GJ81MQez

Setting:

$$F(x)=x\log(x)+\sum _{j=2}^k -|A(x,j)| \tag{1}$$

appears to give the asymptotic $\sqrt{8x\log(x)}$ for the least $k$ such that $F(x)$ is negative.

In general it appears that the least $k$ such that:

$$F(x)=f(x)+\sum _{j=2}^k -|A(x,j)| \tag{2}$$

is negative, has the asymptotic: $\sqrt{8f(x)}$.

Mathematica program at pastebin: https://pastebin.com/GJ81MQez

Setting:

$$F(x)=\frac{x\log(x)}{8}+\sum _{j=2}^k -|A(x,j)| \tag{1}$$

appears to give the asymptotic $\sqrt{x\log(x)}$ for the least $k$ such that $F(x)$ is negative.

In general it appears that the least $k$ such that:

$$F(x)=\frac{f(x)}{8}+\sum _{j=2}^k -|A(x,j)| \tag{2}$$

is negative, has the asymptotic: $\sqrt{f(x)}$.

(*start*)
(*Mathematica*)
Clear[a, f, p];
nn = 2000;
p = 1;
f[n_] := n*Log[n]^p;
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]/8, {n, 1, nn}]];
TableForm[B];
TableForm[B1 = Sign[Transpose[Accumulate[Transpose[B]]]]];
Clear[B];
Quiet[Show[
  ListLinePlot[
   v = ReplaceAll[
     Flatten[Table[First[Position[B1[[n]], -1]], {n, 1, nn}]], 
     First[{}] -> 1], PlotStyle -> Blue], 
  Plot[Sqrt[f[n]], {n, 1, nn}, PlotStyle -> {Red, Thick}], 
  ImageSize -> Large]]
ListLinePlot[v/Table[Sqrt[f[n]], {n, 1, nn}]]
(*end*)

Setting:

$$F(x)=\frac{x\log(x)}{8}+\sum _{j=2}^k -|A(x,j)| \tag{1}$$

appears to give the asymptotic $\sqrt{x\log(x)}$ for the least $k$ such that $F(x)$ is negative.

In general it appears that the least $k$ such that:

$$F(x)=\frac{f(x)}{8}+\sum _{j=2}^k -|A(x,j)| \tag{2}$$

is negative, has the asymptotic: $\sqrt{f(x)}$.

Mathematica program at pastebin: https://pastebin.com/GJ81MQez