Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$.

Probability a number is $(b,1)$-smooth is governed by the Dickman function while probability of number of prime divisors being at least $d$ is governed by the Erdos-Kac formulation.

  1. Is there a known distribution for $(b,c)$-smooth integers?

  2. If we want the joint distribution of $(b,c)$-smooth integers with at least $d$ prime divisors then is there a formulation?

The reason I am asking is it seems we cannot have independence of smoothness and number of prime divisors since it quite does not make sense if the number of divisors is between $\log\log n+0.1\sqrt{\log\log n}$ and $\log\log n+0.2\sqrt{\log\log n}$ then the number cannot be $(b,1)$-smooth where $b=o(f(n))$ at a suitable function $f(n)$ with probability $1-o(1)$ and I do not know to capture this and identify $f(n)$ with independence assumption and identify involved probabilities.


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