Consider on the natural number the lexicographic ordering on the prime factorization: If $m = p_1^{a_1}\cdots p_r^{a_r},n = q_1^{b_1}\cdots q_s^{b_s}$ then we define:

$$m \vartriangleleft n :\iff [(p_1,a_1),(p_2,a_2),\cdots,(p_r,a_r)] \prec [(q_1,b_1),(q_2,b_2),\cdots,(q_s,b_s)]$$

where the right hand side $\prec$ is the lexicographic ordering of the two lists, where $(p,a) \prec (q,b) :\iff p < q \text{ or } ( p=q \text{ and } a < b)$ and the list is sorted by usual absolute value: $p_i < p_{i+1}$.

Example:

1. For instance for $n=1,\cdots , 10 $ we get the following sorting:

$$1, 2, 6, 10, 4, 8, 3, 9, 5, 7$$

2)

Examples : sorted by absolute value: $[2, 7, 11, 30, 60, 121]$ lexicographically sorted: $[2, 30, 60, 7, 11, 121]$

[(2, 1)] = [(2, 1)]
[(2, 1)] < [(7, 1)]
[(2, 1)] < [(11, 1)]
[(2, 1)] < [(2, 1), (3, 1), (5, 1)]
[(2, 1)] < [(2, 2), (3, 1), (5, 1)]
[(2, 1)] < [(11, 2)]
[(7, 1)] > [(2, 1)]
[(7, 1)] = [(7, 1)]
[(7, 1)] < [(11, 1)]
[(7, 1)] > [(2, 1), (3, 1), (5, 1)]
[(7, 1)] > [(2, 2), (3, 1), (5, 1)]
[(7, 1)] < [(11, 2)]
[(11, 1)] > [(2, 1)]
[(11, 1)] > [(7, 1)]
[(11, 1)] = [(11, 1)]
[(11, 1)] > [(2, 1), (3, 1), (5, 1)]
[(11, 1)] > [(2, 2), (3, 1), (5, 1)]
[(11, 1)] < [(11, 2)]
[(2, 1), (3, 1), (5, 1)] > [(2, 1)]
[(2, 1), (3, 1), (5, 1)] < [(7, 1)]
[(2, 1), (3, 1), (5, 1)] < [(11, 1)]
[(2, 1), (3, 1), (5, 1)] = [(2, 1), (3, 1), (5, 1)]
[(2, 1), (3, 1), (5, 1)] < [(2, 2), (3, 1), (5, 1)]
[(2, 1), (3, 1), (5, 1)] < [(11, 2)]
[(2, 2), (3, 1), (5, 1)] > [(2, 1)]
[(2, 2), (3, 1), (5, 1)] < [(7, 1)]
[(2, 2), (3, 1), (5, 1)] < [(11, 1)]
[(2, 2), (3, 1), (5, 1)] > [(2, 1), (3, 1), (5, 1)]
[(2, 2), (3, 1), (5, 1)] = [(2, 2), (3, 1), (5, 1)]
[(2, 2), (3, 1), (5, 1)] < [(11, 2)]
[(11, 2)] > [(2, 1)]
[(11, 2)] > [(7, 1)]
[(11, 2)] > [(11, 1)]
[(11, 2)] > [(2, 1), (3, 1), (5, 1)]
[(11, 2)] > [(2, 2), (3, 1), (5, 1)]
[(11, 2)] = [(11, 2)]

I used this sorting to visualize the following kernel Jaccard similarty kernel, where $\Omega$ counts the prime divisors with multiplicity:

$$\frac{\Omega(\gcd(a,b))}{\Omega(\operatorname{lcm}(a,b))}$$

where the entries $n=1,\ldots, N=1400$ of the matrix are sorted by the lexicographic ordering above:

Q: While there seems to be some visual fractal pattern, I am asking myself, how does one quantify / define this fractal pattern?

One idea would be to let $N \rightarrow \infty$ and make the matrix larger, so that one can zoom in but I am not sure how to formalize this idea.

Edit: Here is another image for the kernel $\omega(\gcd(a,b))$ whose Gram matrix of size $n \times n$ has rank $\pi(n)$, where $\omega$ counts the distinct prime divisors of $n$ and $\pi$ is the prime counting function:

Counting the size of the blocks on the main diagonal, we find the following OEIS sequence:

$$0, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 4, 2, 1, 1, 5, 2, 1, 1, 5, 2, 1, 1, 1, 6, 2, 1, 1, 1, 6, 2, 1, 1, 1, 1, 7, 2, 1, 1, 1, 1, 7, 3, 1, 1, 1, 1, 8, 3, 1, 1, 1, 1, 8, 3, 1, 1, 1, 1, 1, 9, 3, 1, 1, 1, 1, 1, 9, 3, 1, 1, 1, 1, 1, 1, 10, 3, 1, 1, 1, 1, 1, 1, 10, 4, 1, 1, 1, 1, 1, 1, 11, 4, 1, 1, 1, 1, 1, 1, 11, 4, 1, 1, 1, 1, 1, 1, 1$$

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