# [Math] Given a finite list of prime factors, what is the fastest way to find all numbers that can be formed from them

combinatoricselementary-number-theoryprime numbers

$$\text{Let} \ S = \{p_1,p_2,p_3,…,p_n\}$$
$$\text{where} \ p_i \in \Bbb P$$

What is the fastest known method method/algorithm to generate all unique numbers through product operation on $S$?

$\text{Ex}: S= \{3,5,2\}$
Soln:
$3\times5 = 15$
$3\times2 = 6$
$3\times5\times2 = 30$
$5\times2 = 10$

Currently, my ideas hover around generating all subsets of $S$, multiplying all the members in each of them and eliminating the duplicates from the list of numbers so generated. This is $O(2^n)$.

If the primes are distinct and repetitions (e.g. $3\times3\times 5$) are forbidden, then going through the whole list of subsets will yield no duplicates, because the fundamental theorem of arithemetic says prime factorizations are unique.