How to generate all multiplicative partitions of a number if I have a list of primes/exponents?

Question

For instance, the number 24 has prime factorization 2^3*3^1, and can be written in the following ways

1*24
2*12
2*2*6
2*3*4
2*2*2*3
3*8
4*6

I may have missed one but you get the idea.

I tried looking into the other thread How to find multiplicative partitions of any integer? but couldn't understand the answers in all honesty.

I don't need anyone to write code for me but rather I could really use some help creating an efficient algorithm for this (probably something recursive?).

I am coding in Python.

Answer 1

Your problem can be condensed into finding all of the partitions of a set, as each factor (prime and composite) can be represented as the product of the elements of a subset that makes up your partition.

I would represent the factors of your number as a list [2, 2, 2, 3] (well, a set). Here are some possible partitions of this list:

• [2] + [2, 2, 3]
• [2, 2] + [2, 3]
• [2] + [2] + [2, 3]
• [3] + [2] + [2, 2]
• ...

If you multiply together each element of each subset, you will get a factor of the original number:

• 2 * 12
• 4 * 6
• 2 * 2 * 6
• 3 * 2 * 4

You might have to add in a special case for 1 * n.

Here's a relevant question: How can I maximally partition a set?

And another relevant link: Generating the Partitions of a Set