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