Add problem 23 solution

project_euler/problem_23/sol1.py

@@ -0,0 +1,51 @@
+"""
+A perfect number is a number for which the sum of its proper divisors is exactly
+equal to the number. For example, the sum of the proper divisors of 28 would be
+1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.
+
+A number n is called deficient if the sum of its proper divisors is less than n
+and it is called abundant if this sum exceeds n.
+
+As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest
+number that can be written as the sum of two abundant numbers is 24. By
+mathematical analysis, it can be shown that all integers greater than 28123
+can be written as the sum of two abundant numbers. However, this upper limit
+cannot be reduced any further by analysis even though it is known that the
+greatest number that cannot be expressed as the sum of two abundant numbers
+is less than this limit.
+
+Find the sum of all the positive integers which cannot be written as the sum
+of two abundant numbers.
+"""
+
+def solution(limit = 28123):
+    """
+    Finds the sum of all the positive integers which cannot be written as
+    the sum of two abundant numbers
+    as described by the statement above.
+
+    >>> solution()
+    4179871
+    """
+    sumDivs = [1] * (limit + 1)
+
+    for i in range(2, int(limit ** 0.5) + 1):
+        sumDivs[i * i] += i
+        for k in range(i + 1, limit // i + 1):
+            sumDivs[k * i] += k + i
+
+    abundants = set()
+    res = 0
+
+    for n in range(1, limit + 1):
+        if sumDivs[n] > n:
+            abundants.add(n)
+
+        if not any((n - a in abundants) for a in abundants):
+            res+=n
+
+    return res
+
+
+if __name__ == "__main__":
+    print(solution())