The following question naturally came up when dealing with 4-rank of certain class groups. In this case I want to inductively deal with some Legendre symbols, and to do so I want my squarefree integers to be "decently" spaced in the sense below.


Is there an absolute constant $C > 0$ such that for all functions $f$ going to infinity and almost all squarefree integers $n = p\_1 \cdot \ldots \cdot p\_r$ we have
$$
\prod\_{i = 1}^k p\_i < p\_{k + 1}^C
$$
for all $f(n) \leq k < r$? What if we ask the question instead for almost all $k$?



Spacing of prime divisors
This is not true for the all $k<r$ problem. Consider random $n$ below $x$, and put $z=\log x$. How many prime factors would a random number have in $[z,z^e]$? This is approximately Poisson with parameter $\sum\_{z <p \le z^e} 1/p \approx 1$. So with positive probability you would find numbers with as many prime factors from this interval as you care to specify, which means that no fixed value of $C$ would work. 


Look in the work of Ford on the multiplication table problem. There are different ways of phrasing conditions of this type; for example recasting it in terms of the number of prime factors up to some point satisfying a ``barrier" condition. Perhaps one of those reformulations would be useful for you. 


