P4240 毒瘤之神的考验
题目
神仙题\(emmm\)
前置
首先有一个很神奇的性质：
\(\varphi(ij)=\dfrac{\varphi(i)\varphi(j)gcd(i,j)}{\varphi(gcd(i,j))}\)
证明：
具体做法
求\(\sum_{i=1}^n\sum_{j=1}^m\varphi(i,j)\)
我们最终是能得到：$$\begin{aligned}
\sum\limits_{i=1}{n}\sum\limits_{j=1}\varphi(ij)=\sum\limits_{T=1}n\sum\limits_{k|T}\mu(\frac{T}{k})\frac{k}{\varphi(k)}\sum\limits_{i=1}\rfloor}\varphi(iT)\sum\limits_{j=1}^{\lfloor\frac{m}{T}\rfloor}\varphi(jT)\
\end{aligned}$$
网上好像很多人的证明都有些错误
证明：
看我们最终得到的式子，多次查询完全在线做，参数这么多时间肯定承受不住
\(\sum\limits_{k|T}\mu(\frac{T}{k})\frac{k}{\varphi(k)}\)设有函数\(F(x)=\sum\limits_{k|x}\mu(\frac{x}{k})\frac{k}{\varphi(k)}\)再普通不过就不多讲了
\(\sum\limits_{i=1}^{\lfloor\frac{n}{T}\rfloor}\varphi(iT)\)参数有两个，设有函数\(G(y,x)=\sum\limits_{i=1}^{x}\varphi(iy)\)
显然\(G(y,x)=G(y,x-1)+\varphi(xy)\)
整个式子设为函数\(S(y,z,x)=\sum\limits_{T=1}^x\sum\limits_{k|T}\mu(\frac{T}{k})\frac{k}{\varphi(k)}\sum\limits_{i=1}^{y}\varphi(iT)\sum\limits_{j=1}^{z}\varphi(jT)\)
显然\(S(y,z,x)=S(y,z,x-1)+F[x]*G(x,y)*G(x,z)\)
\(F(x)(x<=n),G(y,x)(x<=n,y<=B),S(y,z,x)(x<=n,y,z<=B)\)
B是自己取的任意参数，或许你会想\(\frac{n}{T}>B\)怎么办，\(\Rightarrow T<\frac{n}{B}\)，这部分暴力算，后面分块
#include<cstring>
#include<iostream>
#include<cstdio>
#include<string>
#include<algorithm>
#include<vector>
using namespace std;
typedef long long LL;
const int B=35;
const int maxn=1e5+9;
const LL p=998244353;
inline int Read(){
int x(0),f(1); char c=getchar();
while(c<'0'||c>'9'){
if(c=='-') f=-1; c=getchar();
}
while(c>='0'&&c<='9')
x=(x<<3)+(x<<1)+c-'0',c=getchar();
return x*f;
}
bool visit[maxn];
int mu[maxn],phi[maxn],prime[maxn];
int *G[maxn],*S[B+1][B+1],F[maxn];
int inv[maxn];
inline void First(LL N){
mu[1]=phi[1]=inv[1]=1;
int tot(0);
for(int i=2;i<=N;++i){
if(!visit[i]){
prime[++tot]=i,
mu[i]=-1,
phi[i]=i-1;
}
for(int j=1;j<=tot&&i*prime[j]<=N;++j){
visit[i*prime[j]]=true;
if(i%prime[j]==0){
phi[i*prime[j]]=phi[i]*prime[j];
break;
}else{
phi[i*prime[j]]=phi[i]*phi[prime[j]],
mu[i*prime[j]]=-mu[i];
}
}
}
for(int i=2;i<=N;++i)
inv[i]=p-1ll*p/i*inv[p%i]%p;
for(int i=1;i<=N;++i)
for(int j=1;j*i<=N;++j)
F[i*j]=1ll*(1ll*F[i*j]+1ll*i*inv[phi[i]]%p*mu[j]%p)%p;
for(int i=1;i<=N;++i){
G[i]=new int [N/i+1];
G[i][0]=0;
for(int j=1;j<=N/i;++j)
G[i][j]=1ll*(1ll*G[i][j-1]+1ll*phi[j*i])%p;
}
for(int j=1;j<=B;++j)
for(int k=1;k<=B;++k){
int len(N/(max(j,k)));
S[j][k]=new int [len+1];
S[j][k][0]=0;
for(int i=1;i<=len;++i)
S[j][k][i]=1ll*(1ll*S[j][k][i-1]+1ll*F[i]*G[i][j]%p*G[i][k]%p)%p;
}
}
inline LL Solve(int n,int m){
if(n>m)
swap(n,m);
LL ans(0);
for(int i=1;i<=m/B;++i)
ans=(ans+1ll*F[i]*G[i][n/i]%p*G[i][m/i]%p)%p;
for(int l=m/B+1,r;l<=n;l=r+1){
r=min(n/(n/l),m/(m/l));
ans=(ans+1ll*(S[n/l][m/l][r]-S[n/l][m/l][l-1]+p)%p)%p;
}
return ans;
}
int main(){
int T=Read();
First(100000);
while(T--){
int n(Read()),m(Read());
printf("%lld\n",Solve(n,m));
}
return 0;
}/*
3
1 1
2 2
3 3
1 5 19
*/
