Python/project_euler/problem_078/sol1.py

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"""
Problem 78
Url: https://projecteuler.net/problem=78
Statement:
Let p(n) represent the number of different ways in which n coins
can be separated into piles. For example, five coins can be separated
into piles in exactly seven different ways, so p(5)=7.
OOOOO
OOOO O
OOO OO
OOO O O
OO OO O
OO O O O
O O O O O
Find the least value of n for which p(n) is divisible by one million.
"""
import itertools
def solution(number: int = 1000000) -> int:
"""
>>> solution(1)
1
>>> solution(9)
14
>>> solution()
55374
"""
partitions = [1]
for i in itertools.count(len(partitions)):
item = 0
for j in itertools.count(1):
sign = -1 if j % 2 == 0 else +1
index = (j * j * 3 - j) // 2
if index > i:
break
item += partitions[i - index] * sign
item %= number
index += j
if index > i:
break
item += partitions[i - index] * sign
item %= number
if item == 0:
return i
partitions.append(item)
return 0
if __name__ == "__main__":
import doctest
doctest.testmod()
print(f"{solution() = }")