Is there are way to determine, given a (composite) number $n$ and a list of its prime factors, the next larger (composite) number with the same number of prime factors? Clarification: Not the same number of different factors, but the number of factors overall, including multiplicies.
So for example given $n = 12$ (with the 3 prime factors $2, 2, 3$) I want to find the number $18$ (with the 3 prime factors $2, 3, 3$). And after that, I'd like to get $20$ (with the 3 prime factors $2, 2, 5$).
Put differently I'd like a good way to represent / calculate the monotonically increasing series of numbers with the same number of prime factors.
Assumption: I know the series of prime numbers.