Add solution for the Euler project problem 95. (TheAlgorithms#12669)

* Add documentation and tests for the Euler project problem 95 solution.

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---------

Co-authored-by: Maxim Smolskiy <[email protected]>
Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

project_euler/problem_095/sol1.py

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+"""
+Project Euler Problem 95: https://projecteuler.net/problem=95
+
+Amicable Chains
+
+The proper divisors of a number are all the divisors excluding the number itself.
+For example, the proper divisors of 28 are 1, 2, 4, 7, and 14.
+As the sum of these divisors is equal to 28, we call it a perfect number.
+
+Interestingly the sum of the proper divisors of 220 is 284 and
+the sum of the proper divisors of 284 is 220, forming a chain of two numbers.
+For this reason, 220 and 284 are called an amicable pair.
+
+Perhaps less well known are longer chains.
+For example, starting with 12496, we form a chain of five numbers:
+    12496 -> 14288 -> 15472 -> 14536 -> 14264 (-> 12496 -> ...)
+
+Since this chain returns to its starting point, it is called an amicable chain.
+
+Find the smallest member of the longest amicable chain with
+no element exceeding one million.
+
+Solution is doing the following:
+- Get relevant prime numbers
+- Iterate over product combination of prime numbers to generate all non-prime
+  numbers up to max number, by keeping track of prime factors
+- Calculate the sum of factors for each number
+- Iterate over found some factors to find longest chain
+"""
+
+from math import isqrt
+
+
+def generate_primes(max_num: int) -> list[int]:
+    """
+    Calculates the list of primes up to and including `max_num`.
+
+    >>> generate_primes(6)
+    [2, 3, 5]
+    """
+    are_primes = [True] * (max_num + 1)
+    are_primes[0] = are_primes[1] = False
+    for i in range(2, isqrt(max_num) + 1):
+        if are_primes[i]:
+            for j in range(i * i, max_num + 1, i):
+                are_primes[j] = False
+
+    return [prime for prime, is_prime in enumerate(are_primes) if is_prime]
+
+
+def multiply(
+    chain: list[int],
+    primes: list[int],
+    min_prime_idx: int,
+    prev_num: int,
+    max_num: int,
+    prev_sum: int,
+    primes_degrees: dict[int, int],
+) -> None:
+    """
+    Run over all prime combinations to generate non-prime numbers.
+
+    >>> chain = [0] * 3
+    >>> primes_degrees = {}
+    >>> multiply(
+    ...     chain=chain,
+    ...     primes=[2],
+    ...     min_prime_idx=0,
+    ...     prev_num=1,
+    ...     max_num=2,
+    ...     prev_sum=0,
+    ...     primes_degrees=primes_degrees,
+    ... )
+    >>> chain
+    [0, 0, 1]
+    >>> primes_degrees
+    {2: 1}
+    """
+
+    min_prime = primes[min_prime_idx]
+    num = prev_num * min_prime
+
+    min_prime_degree = primes_degrees.get(min_prime, 0)
+    min_prime_degree += 1
+    primes_degrees[min_prime] = min_prime_degree
+
+    new_sum = prev_sum * min_prime + (prev_sum + prev_num) * (min_prime - 1) // (
+        min_prime**min_prime_degree - 1
+    )
+    chain[num] = new_sum
+
+    for prime_idx in range(min_prime_idx, len(primes)):
+        if primes[prime_idx] * num > max_num:
+            break
+
+        multiply(
+            chain=chain,
+            primes=primes,
+            min_prime_idx=prime_idx,
+            prev_num=num,
+            max_num=max_num,
+            prev_sum=new_sum,
+            primes_degrees=primes_degrees.copy(),
+        )
+
+
+def find_longest_chain(chain: list[int], max_num: int) -> int:
+    """
+    Finds the smallest element of longest chain
+
+    >>> find_longest_chain(chain=[0, 0, 0, 0, 0, 0, 6], max_num=6)
+    6
+    """
+
+    max_len = 0
+    min_elem = 0
+    for start in range(2, len(chain)):
+        visited = {start}
+        elem = chain[start]
+        length = 1
+
+        while elem > 1 and elem <= max_num and elem not in visited:
+            visited.add(elem)
+            elem = chain[elem]
+            length += 1
+
+        if elem == start and length > max_len:
+            max_len = length
+            min_elem = start
+
+    return min_elem
+
+
+def solution(max_num: int = 1000000) -> int:
+    """
+    Runs the calculation for numbers <= `max_num`.
+
+    >>> solution(10)
+    6
+    >>> solution(200000)
+    12496
+    """
+
+    primes = generate_primes(max_num)
+    chain = [0] * (max_num + 1)
+    for prime_idx, prime in enumerate(primes):
+        if prime**2 > max_num:
+            break
+
+        multiply(
+            chain=chain,
+            primes=primes,
+            min_prime_idx=prime_idx,
+            prev_num=1,
+            max_num=max_num,
+            prev_sum=0,
+            primes_degrees={},
+        )
+
+    return find_longest_chain(chain=chain, max_num=max_num)
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")