(This is based on an earlier MSE posting, "Structures in the plot of the “squareness” of numbers.")

My main question is to explain the structural features of this plot:

This is a plot of what I call the

*squareness ratio*$r(n)$ of a natural number $n$ (or simply the "squareness"). The squareness $r(n)$ is the largest ratio $\le 1$ that can be obtained by partitioning the factors of $n$ into two parts and forming the ratio of their products. A perfect square has squareness $1$. A prime $p$ has squareness $1/p$. In a sense, the squareness measures how close is $n$ to a perfect square.

The squareness ratios for the first ten number $n=1,\ldots,10$ are $$1,\frac{1}{2},\frac{1}{3} ,1,\frac{1}{5},\frac{2}{3},\frac{1}{7},\frac{1}{2},1,\frac {2}{5} \;.$$ A more substantive example is $n=12600=2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7$, for which the largest ratio is $$\frac{3 \cdot 5 \cdot 7}{2^3 \cdot 3 \cdot 5}=\frac{7}{8}=0.875 \;.$$ One more example: $n = 2^2 \cdot 3^2 \cdot 5^2 \cdot 7 \cdot 11 = 69300$, $$r(n) = \frac{2^2 \cdot 3^2 \cdot 7}{5^2 \cdot 11}=\frac{252}{275} \approx 0.92 \;.$$

The only feature of this plot—the rays through the origin—that is evident to me is explained by the fact that, for a prime $p$, for $n = k p$ and $k=1,\ldots,p$, the squareness ratio is $k/p$, and so those ratios lie on a line through the origin of slope $\frac{1}{p^2}$. MSE user PattuX emphasized that similar rays occur for particular composite $n$.

Several other features could use explanation:

(1) The discernable density change at $r=\frac{1}{2}$.

(2) The (apparent) hyperbolas.

(3) The interesting structure near $r=1$, both negative (hole-)curves and positive (dot-)curves:

^{ Detail: $35 K \le n \le 60K$ (approximately), near $r=1$. }

I welcome explanations for (1), (2), (3), and other apparent features of the plot. This is to satisfy curiosity; it is far from my expertise.

**: As per Gerhard Paseman's request, the plot with only odd $n$ ratios:**

*Added(1)*^{ Squareness ratio $r(n)$ for odd $n$ only; even $n$ not plotted. }

** Added(2)**: The landscape is rather different for larger $n$
(in accordance with Lucia's insights):

^{ Squareness ratio $r(n)$ for $900{,}000 \le n \le 1{,}000{,}000$. }

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