Added solution for Project Euler problem 77.

project_euler/problem_077/sol1.py

@@ -0,0 +1,66 @@
+"""
+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?
+"""
+
+primes = set(range(3, 100, 2))
+primes.add(2)
+for i in range(3, 100, 2):
+    if i not in primes:
+        continue
+    primes.difference_update(set(range(i * i, 100, i)))
+
+CACHE_PARTITION = {0: {1}}
+
+
+def partition(n: int) -> set:
+    """
+    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}
+    """
+    if n < 0:
+        return set()
+    if n in CACHE_PARTITION:
+        return CACHE_PARTITION[n]
+
+    ret = set()
+    for prime in primes:
+        if prime > n:
+            continue
+        for sub in partition(n - prime):
+            ret.add(sub * prime)
+
+    CACHE_PARTITION[n] = ret
+    return ret
+
+
+def solution(m: int = 5000) -> int:
+    """
+    Return the smallest integer that can be written as the sum of primes in over
+    m unique ways.
+    >>> solution(500)
+    45
+    """
+    for n in range(1, 100):
+        if len(partition(n)) > m:
+            return n
+    return 0
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")