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: $\sqrt{8f(x)}$.

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