crash的数字表格

所谓莫比乌斯反演，就是把好看的式子变得越来越猥琐（题外话）

开始变形：

[sumlimits_{i=1}^nsumlimits_{j=1}^mlca(i,j) ]

[=sumlimits_{i=1}^nsumlimits_{j=1}^mfrac{ij}{gcd(i,j)} ]

[=sumlimits_{d=1}^nfrac1dsumlimits_{i=1}^{left[frac nd ight]}sumlimits_{j=1}^{left[frac md ight]}ijd^2*[gcd(i,j)=1] ]

[=sumlimits_{d=1}^ndsumlimits_{i=1}^{left[frac nd ight]}sumlimits_{j=1}^{left[frac md ight]}ijsumlimits_{k|gcd(i,j)}mu(k) ]

[=sumlimits_{d=1}^ndsumlimits_{k=1}^{left[frac nd ight]}mu(k)k^2sumlimits_{i=1}^{left[frac {n}{dk} ight]}sumlimits_{j=1}^{left[frac {m}{dk} ight]}ij ]

[=sumlimits_{d=1}^ndsumlimits_{k=1}^{left[frac nd ight]}mu(k)k^2frac14left[frac {n}{dk} ight]left(left[frac {n}{dk} ight]+1 ight)left[frac {m}{dk} ight]left(left[frac {m}{dk} ight]+1 ight) ]

这时使出我们的必杀技：令(t=dk)

[frac14sumlimits_{t=1}^nleft[frac {n}{t} ight]left(left[frac {n}{t} ight]+1 ight)left[frac {m}{t} ight]left(left[frac {m}{t} ight]+1 ight)sumlimits_{d|t}mu(d)d^2frac td ]

[=frac14sumlimits_{t=1}^nleft[frac {n}{t} ight]left(left[frac {n}{t} ight]+1 ight)left[frac {m}{t} ight]left(left[frac {m}{t} ight]+1 ight)sumlimits_{d|t}mu(d)dt ]

对于每一个(t)，(sumlimits_{d|t}mu(d)dt)可以预处理，其余分块即可。

附上代码，放心食用：

#include<bits/stdc++.h>
#define int long long
const int maxn=1e7;
const int mod=20101009;
int mu[maxn+10],h[maxn+10],pri[maxn/100],tot;
int n,m;
inline void init(){
	mu[1]=1;
	for(int i=2;i<=n;i++){
		if(!h[i])mu[i]=-1,pri[++tot]=i;
		for(int j=1;j<=tot&&pri[j]*i<=n;j++){
			h[pri[j]*i]=1;
			if(i%pri[j]==0)break;
			mu[pri[j]*i]=-mu[i];
		}
	}
	memset(h,0,sizeof(h));
	for(int i=1;i<=n;i++)
		if(mu[i]!=0)
			for(int j=1;j<=n/i;j++)
				h[i*j]=(h[i*j]+mu[i]*i)%mod;
	for(int i=1;i<=n;i++)
		h[i]=(i*h[i]+h[i-1])%mod;
}
signed main(){
	scanf("%lld%lld",&n,&m);
	if(n>m)std::swap(n,m);
	init();
	int ans=0;
	for(int l=1,r;l<=n;l=r+1){
		r=std::min(n/(n/l),m/(m/l));
		ans=(ans+((((n/l)*(n/l+1)/2)%mod)*(((m/l)*(m/l+1)/2)%mod)%mod*(h[r]-h[l-1])%mod))%mod;
	}
	printf("%lld
",(ans%mod+mod)%mod);
	return 0;
}

深深地感到自己的弱小。