Exploring prime factorization disorder as a signal for nearby primes

About prime factorization of consecutive integers, we all can notice prime factors vary apparently without any logic.

Some numbers like $82 = 2 \times 41$ have highly unequal factors (high variance among the factors), while others like $80 = 2^4 \times 5$ or $2310 = 2 \times 3 \times 5 \times 7 \times 11$ have a stronger balance among the factors (low variance of prime factors). Prime numbers themselves are the extreme case: no factorization at all, so pure order. The powers of primes are another instance of order, but a little more "complicated" (due to the multiplicity).

This makes me wonder whether there is a measurable sense in which primes tend to appear near integers whose factorization is especially disordered, as if the primes act like a release valve, restoring order after a stretch of structural complexity.

Parameters I am already thinking about

For each $n$ (positive integer), I would like to measure something like:

• the variance of the prime factors (with or without multiplicity);
• the number of distinct prime factors and the total count with multiplicity;
• low/high or very unequal multiplicities among the factors.

Question: are there other quantities related to the prime factorization of $n$ (like its factors, their multiplicities, or combinations thereof) that would be interesting to include in an experimental study like this?

I am looking for quantities that might reveal patterns in how primes distribute relative to integers with particular factorization shapes, ideally quantities that behave in a scale-invariant way, so the pattern would look similar (if made possible by the mathematics that underlies everything) whether I am studying numbers near $n = 100$ or near $n = 10^{6}$ and beyond.

I’m not asking for proofs, just suggestions for promising quantities to examine from a computational perspective, and perhaps some insight into why they might (or might not) be related to the proximity of a prime number to the consecutive integers (of varying sizes) I’m analyzing by calculating those quantities.

I've read some related answers in this site, like for example this one, and I would like to delve deeper into the matter but with an experimental approach.