Given a list of (small) primes $ (p_0, p_1, \dots, p_{n-1})$, is there an (efficient) algorithm to enumerate, in order, all numbers that can be expressed as $ \prod_{k=0}^{n-1} p_k^{e_k} $, where $e_k \in \mathbb{Z}, e_k \ge 0 $? What about in a certain interval, potentially at an exponential starting point?
For example, if we had the set $(2,3,5)$, the first few numbers would be $(2, 3, 2^2, 5, 2 \cdot 3, 2^3, 3^2, 2 \cdot 5, \dots )$.
Is there an algorithm to efficiently enumerate all the numbers not expressible as a product of powers of primes from the set? How about in an interval?
Note: I just saw the Polymath paper on deterministic prime finding in an interval ( Deterministic methods to find primes ) and thats what inspired this question. I don't know if it's important that the set be a list of primes, but I'll keep it in there just in case.
EDIT: I was unclear by what I meant by 'efficient' . Let me try making it more precise:
Given a list of $n$ primes $(p_0, p_1, \dots p_{n-1})$ and a bound, $B$, is it possible to find in polynomial time with respect to $lg(B)$ and $n$, the next integer, $x$, such that $x > B$ and is expressible as a product of powers of primes from the list?