質數篩選——素數篩選法

質數篩選

最終算法模板，比賽就寫這個！！

int prime[100010];

void Prime(){
    for (int i = 2; i <= 1000000; i++) {
        prime[i] = true;
    }
    for (int i = 1; i * i <= 1000000; i++) {
        if (prime[i]) {
            for (int j = i * i; j <= 1000000; j += i) {
                prime[j] = false;
            }
        }
    }
}

或者寫下面這個:線性篩

#define SIZE 1000000
int main()
{
    int check[SIZE] = {0};//元素值為0代表是素數
    int prime[SIZE] = {0};
    int pos=0;
    int flag;
    for (int i = 2 ; i < SIZE ; i++)
    {
        if (!check[i])//如果是素數
            prime[pos++] = i;

        for (int j = 0 ; j < pos && i*prime[j] < SIZE ; j++)
        {
            check[i*prime[j]] = 1;//篩掉
            //標注一
            if (i % prime[j] == 0)
                break;
        }
    }

    printf("%.2f", (double)clock()/CLOCKS_PER_SEC);

    return 0;
}

下面是具體解釋，可以不看，比賽套上面模板即可！！！

![](https://img2018.cnblogs.com/blog/1454456/201811/1454456-20181119214228044-1809688019.png)
###如何判斷一個數是不是質數？簡單方法一：
![](https://img2018.cnblogs.com/blog/1454456/201811/1454456-20181119215344441-1163614886.png)

    int is_prime(int n) {
        for (int i = 2; i * i <= n; ++i) {
            if (n % i == 0) {
                return 0; // 不是質數
            }
        }
        return 1; // 是質數
    }

##素數篩選算法：
![](https://img2018.cnblogs.com/blog/1454456/201811/1454456-20181119220344796-975079526.png)

####算法一：時間復雜度---- O(nlogn)
![](https://img2018.cnblogs.com/blog/1454456/201811/1454456-20181119220352279-1219400795.png)

	for (int i = 2; i <= n; ++i) {
	    is_prime[i] = 1;
	}
	for (int i = 2; i <= n; ++i) {
	    for (int j = i * 2; j <= n; j += i) {
	        is_prime[j] = 0;
	    }
	}

####算法二優化：
![](https://img2018.cnblogs.com/blog/1454456/201811/1454456-20181119220423117-1867892215.png)

	for (int i = 2; i <= n; ++i) {
    	    is_prime[i] = 1;
	}
	for (int i = 2; i * i <= n; ++i) {
    	    if (is_prime[i]) {
        	for (int j = i * i; j <= n; j +=i) {
           	  is_prime[j] = 0;
        	}
   	     }
	}