"""

Factor Combinations

Given an integer n, return all possible combinations of its factors.

Factors should be greater than 1 and less than n.

Reference: https://leetcode.com/problems/factor-combinations/

Complexity:

Time: O(n * log(n)) approximate

Space: O(log(n)) recursion depth

"""

from __future__ import annotations

def get_factors(number: int) -> list[list[int]]:

"""Return all factor combinations of number using iteration.

Args:

number: A positive integer.

Returns:

A list of lists, each containing a valid factorization.

Examples:

>>> get_factors(12)

[[2, 6], [2, 2, 3], [3, 4]]

"""

todo: list[tuple[int, int, list[int]]] = [(number, 2, [])]

combinations: list[list[int]] = []

while todo:

remaining, divisor, partial = todo.pop()

while divisor * divisor <= remaining:

if remaining % divisor == 0:

combinations.append(partial + [divisor, remaining // divisor])

todo.append((remaining // divisor, divisor, partial + [divisor]))

divisor += 1

return combinations

def recursive_get_factors(number: int) -> list[list[int]]:

"""Return all factor combinations of number using recursion.

Args:

number: A positive integer.

Returns:

A list of lists, each containing a valid factorization.

Examples:

>>> recursive_get_factors(12)

[[2, 6], [2, 2, 3], [3, 4]]

"""

def _factor(

remaining: int,

divisor: int,

partial: list[int],

combinations: list[list[int]],

) -> list[list[int]]:

while divisor * divisor <= remaining:

if remaining % divisor == 0:

combinations.append(partial + [divisor, remaining // divisor])

_factor(

remaining // divisor, divisor, partial + [divisor], combinations

)

divisor += 1

return combinations

return _factor(number, 2, [], [])