Probability of being B smooth Given an integer N and a smooth base B; what is the (approximate) probability that N is completely divisible by primes <= B.
I assume there is some nice connection to the de Bruijn or Dickman function but I can't see it.
 A: Let $p_1, ... p_n$ be the first $n$ primes and let $\pi_n(N)$ be the number of positive integers less than or equal to $N$ divisible by only the primes $p_i$. This is precisely the number of non-negative integer solutions to
$$\sum_{i=1}^n e_i \log p_i \le \log N$$
which defines an $n$-dimensional region with volume $V = \frac{(\log N)^n}{n! \log p_1 ... \log p_n}$. So this is a first-order estimate of the number of solutions; in other words, asymptotically we have
$$\pi_n(N) \sim \frac{(\log N)^n}{n! \log p_1 ... \log p_n}.$$
(The error term is $O(( \log N)^{n-1})$.) A reasonable guess for the probability that $N$ itself is divisible only by the primes $p_i$ is then roughly the derivative of this, or 
$$\frac{n (\log N)^{n-1}}{N n! \log p_1 ... \log p_n}.$$
A: See "Prime Numbers: A Computational Perspective," by Crandall and Pomerance. The probability that a uniform integer in the ranger ${1, ..., N}$ is $B$-smooth is given there as $u^{-u}$, where $u = \ln N/\ln B$. (I am not sure whether this is heuristic or provable.)
