> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/number-theory/sieve-linear.md).

# 线性筛法（欧拉筛）

### 线性筛（欧拉筛）

每个合数只被它的**最小质因子**筛掉一次，因此复杂度是严格的 **O(n)**。

```cpp
const int N = 1000000;
int primes[N], cnt = 0;
bool notPrime[N + 1];

void linearSieve(int n) {
    for (int i = 2; i <= n; i++) {
        if (!notPrime[i]) primes[cnt++] = i;      // i 是素数
        for (int j = 0; j < cnt && i * primes[j] <= n; j++) {
            notPrime[i * primes[j]] = true;
            if (i % primes[j] == 0) break;        // 关键：保证只被最小质因子筛一次
        }
    }
}
```

### 为什么是 O(n)

每个合数只会被“最小质因子 × 剩余”标记一次，`if (i % primes[j] == 0) break;` 是核心。

### 与埃氏筛对比

|      | 埃氏筛            | 线性筛          |
| ---- | -------------- | ------------ |
| 复杂度  | O(n log log n) | O(n)         |
| 重复标记 | 有              | 无            |
| 用途   | 求素数表           | 求素数表 + 最小质因子 |

线性筛还能顺便求出每个数的**最小质因子**，用于质因数分解。

### 应用

* 预处理素数表（配合二分、数论题）
* 求约数个数、欧拉函数等积性函数


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