[BZOJ 2693]jzptab：莫比乌斯反演

/*
User:Small
Language:C++
Problem No.:2693
*/
#include<bits/stdc++.h>
#define ll long long
#define inf 999999999
using namespace std;
const int M=1e7+5;
const ll mod=1e8+9;
int n,m,mu[M],prime[M],cnt;
ll ans,h[M],sum[M];
bool not_prime[M];
ll getsum(int n,int m){
n=(ll)n*(n+1)/2%mod;
m=(ll)m*(m+1)/2%mod;
return (ll)n*m%mod;
}
void solve(){
ans=0;
scanf("%d%d",&n,&m);
if(n>m) swap(n,m);
for(int i=1,r;i<=n;i=r+1){
r=min(n/(n/i),m/(m/i));
ans=(ans+(sum[r]-sum[i-1]+mod)%mod*getsum(n/i,m/i)%mod)%mod;
}
printf("%lld\n",(ans+mod)%mod);
}
int main(){
freopen("data.in","r",stdin);//
mu[1]=h[1]=1;
for(int i=2;i<=1e7;i++){
if(!not_prime[i]){
prime[++cnt]=i;
mu[i]=-1;
h[i]=(i-(ll)i*i)%mod;
}
for(int j=1;j<=cnt&&i*prime[j]<=1e7;j++){
not_prime[i*prime[j]]=1;
if(i%prime[j]==0){
mu[i*prime[j]]=0;
h[i*prime[j]]=h[i]*prime[j]%mod;
break;
}
mu[i*prime[j]]=-mu[i];
h[i*prime[j]]=h[i]*h[prime[j]]%mod;
}
}
for(int i=1;i<=1e7;i++) sum[i]=(sum[i-1]+h[i])%mod;
int t;
scanf("%d",&t);
while(t--) solve();
return 0;
}