# Smooth integers with lower bound on $\omega(n)$

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.