Heres the initial upgraded number fingerprinter I talked about in the past and some results and an explanation below.

Note that these are wide black images on ibb, so they appear as a tall thin strip near the top of ibb as if they're part of the website. They practically blend in. Right click the blackstrip and hit 'view image' and then zoom in.

https://ibb.co/26JmZXB
https://ibb.co/LpJpggq
https://ibb.co/Jt2Hsgt
https://ibb.co/hcxrFfV
https://ibb.co/BKZNzng
https://ibb.co/L6BtXZ4
https://ibb.co/yVHZNq4
https://ibb.co/tQXS8Hr

https://paste.ofcode.org/an4LcpkaKr...

Hastebin wouldn't save for some reason so paste.ofcode.org it is.

Not much to look at, but I was thinking I'd maybe mark the columns where gaps occur and do some statistical tests like finding the stds of the gaps, density, etc. The type test I wrote categorizes products into 11 different types, based on the value of a subset of variables taken from a vector of a couple hundred variables but I didn't want to include all that mess of code. And I was thinking of maybe running this fingerprinter on a per type basis, set to repeat, and looking for matching indexs (pixels) to see what products have in common per type.

Or maybe using them to train a classifier of some sort.

Each fingerprint of a product shares something like 16-20% of indexes with it's factors, so I'm thinking thats an avenue to explore.

What the fingerprinter does is better explained by the subfunction findAb.

The code contains a comment explaining this, but basically the function destructures a number into a series of division and subtractions, and makes a note of how many divisions in a 'run'.
Typically this is for numbers divisible by 2.

So a number like 35 might look like this, when done
p = 35
((((p-1)/2)-1)/2/2/2/2)-1

And we'd represent that as

ab(w, x, y, z)
Where w is the starting value 35 in this case,
x is the number to divide by at each step, y is the adjustment (how much to subtract by when we encounter a number not divisible by x), and z is a string or vector of our results

which looks something like

ab(35, 2, 1, [1, 4])

Why [1,4]
because we were only able to divide by 2 once, before having to subtract 1, and repeat the process. And then we had a run of 4 divisions.

And for the fingerprinter, we do this for each prime under our number p, the list returned becoming another row in our fingerprint. And then that gets converted into an image.

And again, what I find interesting is that
unknown factors of products appear to share many of these same indexes.

What I might do is for, each individual run of Ab, I might have some sort of indicator for when *another* factor is present in the current factor list for each index. So I might ask, at the given step, is the current result (derived from p), divisible by 2 *and* say, 3? If so, mark it.

And then when I run this through the fingerprinter itself, all those pixels might get marked by a different color, say, make them blue, or vary their intensity based on the number of factors present, I don't know. Whatever helps the untrained eye to pick up on leads, clues, and patterns.

If it doesn't make sense, take another look at the example:
((((p-1)/2)-1)/2/2/2/2)-1

This is semi-unique to each product. After the fact, you can remove the variable itself, and keep just the structure in question, replacing the first variable with some other number, and you get to see what pops out the otherside.

If it helps, you can think of the structure surrounding our variable p as the 'electron shell', the '-1's as bandgaps, and the runs of '2's as orbitals, with the variable at the center acting as the 'nucleus', with the factors of that nucleus acting as the protons and neutrons, or nougaty center lol.

Anyway I just wanted to share todays flavor of insanity on the off chance someone might enjoy reading it.