BZOJ 1968: [Ahoi2005]COMMON 约数研究

1968

线性筛求约数个数

一般跟质因子或者质因子个数有关的积性函数都可以用线性筛求

比如欧拉函数、莫比乌斯反演函数、约数个数函数、约数和函数等函数

考虑最小的质因子对转移的影响

代码：

#pragma GCC optimize(2)
#pragma GCC optimize(3)
#pragma GCC optimize(4)
#include<bits/stdc++.h>
using namespace std;
#define fi first
#define se second
#define pi acos(-1.0)
#define LL long long
//#define mp make_pair
#define pb push_back
#define ls rt<<1, l, m
#define rs rt<<1|1, m+1, r
#define ULL unsigned LL
#define pll pair<LL, LL>
#define pli pair<LL, int>
#define pii pair<int, int>
#define piii pair<pii, int>
#define pdd pair<long double, long double>
#define mem(a, b) memset(a, b, sizeof(a))
#define fio ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#define fopen freopen("in.txt", "r", stdin);freopen("out.txt", "w", stout);
//head

const int N = 1e6 + 5;
int f[N], num[N], prime[N];
bool not_p[N];
void seive(int n) {
    int tot = 0;
    f[1] = 1;
    for (int i = 2; i <= n; i++) {
        if(!not_p[i]) {
            prime[++tot] = i;
            f[i] = 2;
            num[i] = 1;
        }
        for (int j = 1; i*prime[j] <= n; j++) {
            not_p[i*prime[j]] = true;
            if(i%prime[j]) {
                f[i*prime[j]] = f[i]*2;
                num[i*prime[j]] = 1;
            }
            else {
                f[i*prime[j]] = f[i]/(num[i]+1)*(num[i]+2);
                num[i*prime[j]] = num[i] + 1;
                break;
            }
        }
    }
    LL ans = 1;
    for (int i = 2; i <= n; i++) ans += f[i];
    printf("%lld\n", ans);
}
int main() {
    int n;
    scanf("%d", &n);
    seive(n);
    return 0;
}