|
| 1 | +--- |
| 2 | +title: 762.二进制表示中质数个计算置位:位运算(mask O(1)判断) |
| 3 | +date: 2026-02-21 12:30:01 |
| 4 | +tags: [题解, LeetCode, 简单, 位运算, 数学, 质数] |
| 5 | +categories: [题解, LeetCode] |
| 6 | +--- |
| 7 | + |
| 8 | +# 【LetMeFly】762.二进制表示中质数个计算置位:位运算(mask O(1)判断) |
| 9 | + |
| 10 | +力扣题目链接:[https://leetcode.cn/problems/prime-number-of-set-bits-in-binary-representation/](https://leetcode.cn/problems/prime-number-of-set-bits-in-binary-representation/) |
| 11 | + |
| 12 | +<p>给你两个整数 <code>left</code> 和 <code>right</code> ,在闭区间 <code>[left, right]</code> 范围内,统计并返回 <strong>计算置位位数为质数</strong> 的整数个数。</p> |
| 13 | + |
| 14 | +<p><strong>计算置位位数</strong> 就是二进制表示中 <code>1</code> 的个数。</p> |
| 15 | + |
| 16 | +<ul> |
| 17 | + <li>例如, <code>21</code> 的二进制表示 <code>10101</code> 有 <code>3</code> 个计算置位。</li> |
| 18 | +</ul> |
| 19 | + |
| 20 | +<p> </p> |
| 21 | + |
| 22 | +<p><strong>示例 1:</strong></p> |
| 23 | + |
| 24 | +<pre> |
| 25 | +<strong>输入:</strong>left = 6, right = 10 |
| 26 | +<strong>输出:</strong>4 |
| 27 | +<strong>解释:</strong> |
| 28 | +6 -> 110 (2 个计算置位,2 是质数) |
| 29 | +7 -> 111 (3 个计算置位,3 是质数) |
| 30 | +9 -> 1001 (2 个计算置位,2 是质数) |
| 31 | +10-> 1010 (2 个计算置位,2 是质数) |
| 32 | +共计 4 个计算置位为质数的数字。 |
| 33 | +</pre> |
| 34 | + |
| 35 | +<p><strong>示例 2:</strong></p> |
| 36 | + |
| 37 | +<pre> |
| 38 | +<strong>输入:</strong>left = 10, right = 15 |
| 39 | +<strong>输出:</strong>5 |
| 40 | +<strong>解释:</strong> |
| 41 | +10 -> 1010 (2 个计算置位, 2 是质数) |
| 42 | +11 -> 1011 (3 个计算置位, 3 是质数) |
| 43 | +12 -> 1100 (2 个计算置位, 2 是质数) |
| 44 | +13 -> 1101 (3 个计算置位, 3 是质数) |
| 45 | +14 -> 1110 (3 个计算置位, 3 是质数) |
| 46 | +15 -> 1111 (4 个计算置位, 4 不是质数) |
| 47 | +共计 5 个计算置位为质数的数字。 |
| 48 | +</pre> |
| 49 | + |
| 50 | +<p> </p> |
| 51 | + |
| 52 | +<p><strong>提示:</strong></p> |
| 53 | + |
| 54 | +<ul> |
| 55 | + <li><code>1 <= left <= right <= 10<sup>6</sup></code></li> |
| 56 | + <li><code>0 <= right - left <= 10<sup>4</sup></code></li> |
| 57 | +</ul> |
| 58 | + |
| 59 | + |
| 60 | + |
| 61 | +## 解题方法:预处理 |
| 62 | + |
| 63 | +写个脚本,计算$10^6$范围内二进制下最多有多少个1:$1^{20}=1048576$ |
| 64 | + |
| 65 | +再算出来其中都有哪些是质数:$[2, 3, 5, 7, 11, 13, 17, 19]$ |
| 66 | + |
| 67 | +再使用一个整数二进制下的每一位代表一个数是否是质数。 |
| 68 | + |
| 69 | +```python |
| 70 | +''' |
| 71 | +LastEditTime: 2026-02-21 12:22:44 |
| 72 | +''' |
| 73 | +max_length = 0 |
| 74 | +while 1 << max_length < 1000000: |
| 75 | + max_length += 1 |
| 76 | +print(f'max_length: {max_length}, 1 << {max_length} = {1 << max_length}') |
| 77 | + |
| 78 | +primes = [] |
| 79 | +for i in range(2, max_length + 1): |
| 80 | + isnot = False |
| 81 | + for j in range(2, i): |
| 82 | + if i % j == 0: |
| 83 | + isnot = True |
| 84 | + break |
| 85 | + if not isnot: |
| 86 | + primes.append(i) |
| 87 | +print(f'primes: {primes}') |
| 88 | + |
| 89 | +mask = 0 |
| 90 | +for p in primes: |
| 91 | + mask |= 1 << p |
| 92 | +print(f'mask: {mask}') |
| 93 | + |
| 94 | +""" |
| 95 | +max_length: 20, 1 << 20 = 1048576 |
| 96 | +primes: [2, 3, 5, 7, 11, 13, 17, 19] |
| 97 | +mask: 665772 |
| 98 | +""" |
| 99 | +``` |
| 100 | + |
| 101 | ++ 时间复杂度$O(right - left)$ |
| 102 | ++ 空间复杂度$O(N\log N)$ |
| 103 | + |
| 104 | +### AC代码 |
| 105 | + |
| 106 | +#### C++ |
| 107 | + |
| 108 | +```cpp |
| 109 | +/* |
| 110 | + * @LastEditTime: 2026-02-21 12:28:09 |
| 111 | + */ |
| 112 | +class Solution { |
| 113 | +private: |
| 114 | + constexpr static int mask = 665772; |
| 115 | +public: |
| 116 | + int countPrimeSetBits(int left, int right) { |
| 117 | + int ans = 0; |
| 118 | + for (int i = left; i <= right; i++) { |
| 119 | + if (mask & (1 << __builtin_popcount(i))) { |
| 120 | + // printf("i = %d, cnt1 = %d\n", i, __builtin_popcount(i)); |
| 121 | + ans++; |
| 122 | + } |
| 123 | + } |
| 124 | + return ans; |
| 125 | + } |
| 126 | +}; |
| 127 | +``` |
| 128 | +
|
| 129 | +> 同步发文于[CSDN](https://letmefly.blog.csdn.net/article/details/158262877)和我的[个人博客](https://blog.letmefly.xyz/),原创不易,转载经作者同意后请附上[原文链接](https://blog.letmefly.xyz/2026/02/21/LeetCode%200762.%E4%BA%8C%E8%BF%9B%E5%88%B6%E8%A1%A8%E7%A4%BA%E4%B8%AD%E8%B4%A8%E6%95%B0%E4%B8%AA%E8%AE%A1%E7%AE%97%E7%BD%AE%E4%BD%8D/)哦~ |
| 130 | +> |
| 131 | +> 千篇源码题解[已开源](https://github.com/LetMeFly666/LeetCode) |
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