Added solution for Project Euler problem 77 (#3132)

* Added solution for Project Euler problem 77.

* Update docstrings, doctest, type annotations and 0-padding in directory name. Reference: #3256

* Implemented lru_cache, better type hints, more doctests for problem 77

* updating DIRECTORY.md

* updating DIRECTORY.md

* Added solution for Project Euler problem 77. Fixes: 2695

* Update docstrings, doctest, type annotations and 0-padding in directory name. Reference: #3256

* Implemented lru_cache, better type hints, more doctests for problem 77

* better variable names

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

Author: fpringle

Diff for: DIRECTORY.md

@@ -709,6 +709,8 @@
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_075/sol1.py)
   * Problem 076
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_076/sol1.py)
+  * Problem 077
+    * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_077/sol1.py)
   * Problem 080
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_080/sol1.py)
   * Problem 081

Diff for: project_euler/problem_077/__init__.py

@@ -0,0 +1,81 @@
+"""
+Project Euler Problem 77: https://projecteuler.net/problem=77
+
+It is possible to write ten as the sum of primes in exactly five different ways:
+
+7 + 3
+5 + 5
+5 + 3 + 2
+3 + 3 + 2 + 2
+2 + 2 + 2 + 2 + 2
+
+What is the first value which can be written as the sum of primes in over
+five thousand different ways?
+"""
+
+from functools import lru_cache
+from math import ceil
+from typing import Optional, Set
+
+NUM_PRIMES = 100
+
+primes = set(range(3, NUM_PRIMES, 2))
+primes.add(2)
+prime: int
+
+for prime in range(3, ceil(NUM_PRIMES ** 0.5), 2):
+    if prime not in primes:
+        continue
+    primes.difference_update(set(range(prime * prime, NUM_PRIMES, prime)))
+
+
+@lru_cache(maxsize=100)
+def partition(number_to_partition: int) -> Set[int]:
+    """
+    Return a set of integers corresponding to unique prime partitions of n.
+    The unique prime partitions can be represented as unique prime decompositions,
+    e.g. (7+3) <-> 7*3 = 12, (3+3+2+2) = 3*3*2*2 = 36
+    >>> partition(10)
+    {32, 36, 21, 25, 30}
+    >>> partition(15)
+    {192, 160, 105, 44, 112, 243, 180, 150, 216, 26, 125, 126}
+    >>> len(partition(20))
+    26
+    """
+    if number_to_partition < 0:
+        return set()
+    elif number_to_partition == 0:
+        return {1}
+
+    ret: Set[int] = set()
+    prime: int
+    sub: int
+
+    for prime in primes:
+        if prime > number_to_partition:
+            continue
+        for sub in partition(number_to_partition - prime):
+            ret.add(sub * prime)
+
+    return ret
+
+
+def solution(number_unique_partitions: int = 5000) -> Optional[int]:
+    """
+    Return the smallest integer that can be written as the sum of primes in over
+    m unique ways.
+    >>> solution(4)
+    10
+    >>> solution(500)
+    45
+    >>> solution(1000)
+    53
+    """
+    for number_to_partition in range(1, NUM_PRIMES):
+        if len(partition(number_to_partition)) > number_unique_partitions:
+            return number_to_partition
+    return None
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")