The attchd graphics are some sample divisor lattices of natural numbers programmed with Mathematica.
Excuse that due to cropping (to fit the Math.SE format), there are fewer columns than the number ranges to left indicate: the label in the first row says 0-23 but the column is cropped so that 12 is the highest number on the first row.
The largest natural number represented is ~100k, with larger ones only limited by computing power and layout issues. But even at this low cardinality, it's interesting to see how the more complex lattices are mixed in with the more basic ones.
Ignoring the lattice labels, it might seem that the distribution of basic divisor lattice parameters may be of interest: like height (max cardinality of a chain in it) and width (max cardinality of an antichain), - after all, primes and prime powers are special cases - and maybe dimension is an independent parameter,
How do number theorists encode the divisor lattice parameters to study distributions? Can it be done via Riemann zeta-type functions? Or completely unrelated approaches?
References would be great if possible.