Symbol(clack:cancel)

solutions/0000_0029/0029.DistinctPowers/euler_29.py

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+"""
+Consider all integer combinations of a^b for 2<=a<=5 and 2<=b<=5 :
+
+2^2 = 4  2^3 = 8  2^4 = 16  2^5 = 32
+3^2 = 9  3^3 = 27  3^4 = 81  3^5 = 243
+4^2 = 16  4^3 = 64  4^4 = 256  4^5 = 1024
+5^2 = 25  5^3 = 125  5^4 = 625  5^5 = 3125
+
+If they are then placed in numerical order, with any repeats removed,
+we get the following sequence of 15 distinct terms:
+
+4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125
+
+How many distinct terms are in the sequence generated by a^b for 2<=a<=100 and 2<=b<=100 ?
+"""
+
+import time
+
+
+def timer(func):
+    def wrapper(*args, **kwargs):
+        start_time = time.time()
+        result = func(*args, **kwargs)
+        end_time = time.time()
+        print(f"{func.__name__} runtime: {end_time - start_time} seconds")
+        return result
+
+    return wrapper
+
+
+def count_distinct_terms_efficient(limit_a, limit_b):
+    # 预计算每个 a 的最小底数表示
+    base_map = {}
+    power_map = {}
+
+    for a in range(2, limit_a + 1):
+        base_map[a] = a
+        power_map[a] = 1
+
+    # 找出所有可以表示为幂的数
+    for power in range(2, 7):  # 2^7=128>100，所以最多到6次幂
+        base = 2
+        while base**power <= limit_a:
+            value = base**power
+            # 如果这个值还没有被更小的底数表示
+            if base_map[value] == value:
+                base_map[value] = base
+                power_map[value] = power
+            base += 1
+
+    # 使用集合存储 (base, exponent) 对
+    seen = set()
+
+    for a in range(2, limit_a + 1):
+        base = base_map[a]
+        power = power_map[a]
+
+        # 如果 a 是某个数的幂，我们需要检查 base 是否还能分解
+        # 例如：16 = 4^2，但 4 = 2^2，所以 16 = 2^4
+        # 我们需要找到最终的 base
+        while base_map[base] != base:
+            power = power * power_map[base]
+            base = base_map[base]
+
+        for b in range(2, limit_b + 1):
+            seen.add((base, power * b))
+
+    return len(seen)
+
+
+def do_compute(a_top: int, b_top: int, a_down: int = 2, b_down: int = 2) -> int:
+    tmp = set()
+    for a in range(a_down, a_top + 1):
+        for b in range(b_down, b_top + 1):
+            tmp.add(a**b)
+    return len(tmp)
+
+
+@timer
+def main():
+    """not bad"""
+    print(count_distinct_terms_efficient(100, 100))
+
+
+@timer
+def main2():
+    print(do_compute(100, 100))
+
+
+if __name__ == "__main__":
+    main()
+    main2()