数论 - 线性筛法与积性函数

首先以求1000000以内的素数为例来探讨筛法

Eratosthenes筛法（埃拉托斯特尼筛法）

时间复杂度：O(N*loglogN)
空间复杂度：O(N)

代码：

#include <map>#include <set>#include <list>#include <cmath>#include <deque>#include <queue>#include <stack>#include <bitset>#include <cctype>#include <cstdio>#include <string>#include <vector>#include <complex>#include <cstdlib>#include <cstring>#include <fstream>#include <sstream>#include <utility>#include <iostream>#include <algorithm>#include <functional>#define LL long long#define INF 0x7fffffffusing namespace std;const int maxn = 1000005;bool vis[maxn];int prime[maxn];int tot;void init() {    tot = 0;    memset(vis, false, sizeof(vis));    for(int i = 2; i < maxn; i ++) {        if(!vis[i]) {            prime[tot ++] = i;            for(int j = i * 2; j < maxn; j += i) {                vis[j] = true;            }        }    }}int main() {    init();    for(int i = 0; i < 100; i ++) {        cout << prime[i] << " ";    }    return 0;}

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Euler筛法（欧拉筛法）

每个合数只会被它最小的质因数筛去，因此时间复杂度为O(N)。

此种为线性筛法

代码：

#include <map>#include <set>#include <list>#include <cmath>#include <deque>#include <queue>#include <stack>#include <bitset>#include <cctype>#include <cstdio>#include <string>#include <vector>#include <complex>#include <cstdlib>#include <cstring>#include <fstream>#include <sstream>#include <utility>#include <iostream>#include <algorithm>#include <functional>#define LL long long#define INF 0x7fffffffusing namespace std;const int maxn = 1000005;bool vis[maxn];int prime[maxn];int tot;void init() {    tot = 0;    memset(vis, false, sizeof(vis));    for(int i = 2; i < maxn; i ++) {        if(!vis[i]) prime[tot ++] = i;        for(int j = 0; j < tot; j ++) {            if(i * prime[j] > maxn) break;            vis[i * prime[j]] = true;            if(i % prime[j] == 0) break;        }    }}int main() {    init();    for(int i = 0; i < 100; i ++) {        cout << prime[i] << " ";    }    return 0;}

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线性筛法求欧拉函数

代码：

#include <map>#include <set>#include <list>#include <cmath>#include <deque>#include <queue>#include <stack>#include <bitset>#include <cctype>#include <cstdio>#include <string>#include <vector>#include <complex>#include <cstdlib>#include <cstring>#include <fstream>#include <sstream>#include <utility>#include <iostream>#include <algorithm>#include <functional>#define LL long long#define INF 0x7fffffffusing namespace std;const int maxn = 1000005;bool vis[maxn];int prime[maxn];int fai[maxn];int tot;void init() {    memset(vis, false, sizeof(vis));    fai[1] = 1;    tot = 0;    for(int i = 2; i < maxn; i ++) {        if(!vis[i]) {            prime[tot ++] = i;            fai[i] = i - 1;        }        for(int j = 0; j < tot; j ++) {            if(i * prime[j] >= maxn) break;            vis[i * prime[j]] = true;            if(i % prime[j] == 0) {                fai[i * prime[j]] = fai[i] * prime[j];                break;            }            else {                fai[i * prime[j]] = fai[i] * (prime[j] - 1);            }        }    }}int main() {    init();    for(int i = 1; i < 100; i ++) {        cout << fai[i] << " ";    }    return 0;}

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线性筛法求莫比乌斯函数

代码：

#include <map>#include <set>#include <list>#include <cmath>#include <deque>#include <queue>#include <stack>#include <bitset>#include <cctype>#include <cstdio>#include <string>#include <vector>#include <complex>#include <cstdlib>#include <cstring>#include <fstream>#include <sstream>#include <utility>#include <iostream>#include <algorithm>#include <functional>#define LL long long#define INF 0x7fffffffusing namespace std;const int maxn = 1000005;bool vis[maxn];int prime[maxn];int mu[maxn];//莫比乌斯函数int tot;void init() {    memset(vis, false, sizeof(vis));    mu[1] = 1;    tot = 0;    for(int i = 2; i < maxn; i ++) {        if(!vis[i]) {            prime[tot ++] = i;            mu[i] = -1;        }        for(int j = 0; j < tot; j ++) {            if(i * prime[j] >= maxn) break;            vis[i * prime[j]] = true;            if(i % prime[j] == 0) {                mu[i * prime[j]] = 0;                break;            }            else {                mu[i * prime[j]] = -mu[i];            }        }    }}int main() {    init();    for(int i = 1; i < 100; i ++) {        cout << mu[i] << " ";    }    return 0;}

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