On divisor-counting ratios and the smoothness of integers

Introduction.
Smoothness, roughness, and divisor structures are central topics in analytic and probabilistic number theory. In previous work, we introduced a ratio, R, associated with non-square integers, defined through a partition of their divisors, and conjectured that the condition R=2 is equivalent to P(n) being greater than the square root of n, where P(n) denotes the largest prime factor of n. Although this conjecture appears simple at first glance, its proof seems highly nontrivial. This difficulty motivates a broader investigation of the behaviour of R beyond the special case R=2.

Methods.
We analyse the general structure and distribution of the ratio R for non-square integers. Our approach combines asymptotic methods, heuristic arguments, and tools from probabilistic number theory, including the Dickman-de Bruijn function, to study how arithmetic properties such as smoothness and the size of the largest prime factor influence the possible values of R.

Results.
We identify several new patterns in the behaviour of R and provide evidence that its distribution reflects deeper structural properties of integer factorisation. In particular, we show that the case R=2 belongs to a broader family of phenomena, and we describe conditions under which other values of R arise.

Conclusions.
This work provides a more comprehensive perspective on the ratio R and highlights its potential as a tool for understanding smoothness and divisor structures. The results open several directions for future research, including a proof of the original conjecture and extensions to related arithmetic functions.