BZOJ 3512: DZY Loves Math IV [杜教筛]

3512: DZY Loves Math IV

题意：求\(\sum_{i=1}^n \sum_{j=1}^m \varphi(ij)\)，\(n \le 10^5, m \le 10^9\)

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n较小，考虑写成前缀和的形式，计算\(S(n,m)=\sum_{i=1}^m \varphi(in)\)

一开始想出

\[n= \prod_i p_i,\ \varphi(in) = \varphi(i) \cdot \varphi(\frac{n}{d})\cdot d,\ d=(n,i) \]

比较好想，共有的质因子应该乘\(p\)而不是\(p-1\)

然后带进去枚举gcd用莫比乌斯反演的套路，中间的函数很奇怪不好算前缀和...

orz了题解，发现题解使用\(\varphi * 1 =id\)来替换

\[\varphi(in) = \varphi(i) \cdot \varphi(\frac{n}{d})\cdot \sum_{e\mid d} \varphi(e) = \varphi(i) \cdot \sum_{e\mid d}\varphi(\frac{n}{e}) \]

因为n是不同质因子的乘积，所以可以把两个\(\varphi\)乘起来

这一步替换和用\(\mu * 1 = \epsilon\)替换有异曲同工之妙，都是将\(gcd\)等于的限制弱化了，变成了整除的关系

推倒后得到

\[S(n,m) = \sum_{d\mid n}\varphi(\frac{n}{d})\cdot S(d, \lfloor \frac{m}{d} \rfloor) \]

对于n不是不同质因子的乘积的，根据\(\varphi\)的公式，多的质因子次数直接提出来乘上就行了

然后记忆化搜索，\(n=1\)就是\(\varphi\)的前缀和

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
#include <map>
using namespace std;
typedef long long ll;
const int N=1664512, U=1664510, mo = 1e9+7;
inline int read(){
    char c=getchar(); int x=0,f=1;
    while(c<'0' || c>'9') {if(c=='-')f=-1; c=getchar();}
    while(c>='0' && c<='9') {x=x*10+c-'0'; c=getchar();}
    return x*f;
}

int n, m;
inline void mod(int &x) {if(x>=mo) x-=mo; else if(x<0) x+=mo;}
bool notp[N]; int p[N/10], phi[N], pr[N];
void sieve(int n) {
	phi[1]=1; pr[1] = 1;
	for(int i=2; i<=n; i++) {
		if(!notp[i]) p[++p[0]] = i, phi[i] = i-1, pr[i] = i;
		for(int j=1; j <= p[0] && i*p[j] <= n; j++) {
			int t = i*p[j];
			notp[ t ] = 1;
			if(i % p[j] == 0) {
				phi[t] = phi[i] * p[j];
				pr[t] = pr[i];
				break;
			}
			phi[t] = phi[i] * (p[j] - 1); 
			pr[t] = pr[i] * p[j];
		}
		mod(phi[i] += phi[i-1]);
	}
}

namespace ha {
	const int p = 1001001;
	struct meow{int ne, val, r;} e[3000];
	int cnt=1, h[p];
	inline void insert(int x, int val) {
		int u = x % p;
		for(int i=h[u];i;i=e[i].ne) if(e[i].r == x) return;
		e[++cnt] = (meow){h[u], val, x}; h[u] = cnt;
	}
	inline int quer(int x) {
		int u = x % p;
		for(int i=h[u];i;i=e[i].ne) if(e[i].r == x) return e[i].val;
		return -1;
	}
} using ha::insert; using ha::quer;

int dj_s(int n) {
	if(n <= U) return phi[n];
	if(quer(n) != -1) return quer(n);
	int ans = (ll) n * (n+1) / 2 %mo, r;
	for(int i=2; i<=n; i=r+1) {
		r = n/(n/i);
		mod(ans -= (ll) dj_s(n/i) * (r-i+1) %mo);
	}
	insert(n, ans);
	return ans;
}

inline int Pow(int a, int b) {
	int ans=1;
	for(; b; b>>=1, a=a*a)
		if(b&1) ans=ans*a;
	return ans;
}
inline ll Phi(int n) {
	int ans = 1;
	if(n <= U) {mod(ans = phi[n] - phi[n-1]); return ans;}
	int m = sqrt(n);
	for(int i=1; p[i] <= m; i++) if(n % p[i] == 0) {
		int a = 0;
		while(n % p[i] == 0) a++, n /= p[i];
		ans *= Pow(p[i], a-1) * (p[i] - 1);
	}
	return ans;
}

map<int, int> Map[N];
int S(int n, int m) {
	if(m == 0) return 0; 
	if(n == 1) return dj_s(m);
	if(Map[n][m]) return Map[n][m];
	//printf("S %d %d\n", n, m);
	int ans = 0;
	for(int i=1; i*i <= n; i++) if(n%i == 0) {
		int j = n/i;
		mod(ans += Phi(j) * S(i, m/i) %mo);
		if(i != j) mod(ans += Phi(i) * S(j, m/j) %mo);
	}
	Map[n][m]=ans;
	return ans;
}
int main() {
	freopen("in", "r", stdin);
	sieve(U);
	n=read(); m=read();
	int ans = 0;
	for(int i=1; i<=n; i++) mod(ans += (ll) i / pr[i] * S(pr[i], m) %mo);
	//for(int i=1; i<=n; i++) printf("nnnnnnnn %d  %d\n", i, S(i, m));
	printf("%d\n", ans);
}