正题

P1829

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题目大意

给出n,m，求$$\sum _{i=1}^n\sum _{j=1}^{m}lcm(i,j)$$

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解题思路

$$\sum _{i=1}^n\sum _{j=1}^{m}lcm(i,j)$$
$$\sum _{d=1}^{n}d\sum _{i=1}^{n/d}\sum _{j=1}^{m/d}i\ast j[gcd(i,j)==1]$$
$$\sum _{d=1}^{n}d\sum _{i=1}^{n/d}\sum _{j=1}^{m/d}i\ast j\sum _{c|i,c|j}\mu (c)$$
$$\sum _{d=1}^{n}d\sum _{c=1}^{n/d}\mu (c)\times c^2\times \sum _{i=1}^{n/cd}i\times \sum _{j=1}^{m/cd}j$$

考虑枚举cd，设k=cd，

$$\sum _{k=1}^n\left(\sum _{i=1}^{n/k}i\times \sum _{j=1}^{m/k}j\times \sum _{d|k}d\mu (\frac{k}{d})(\frac{k}{d}{)}^2\right)$$
$$\sum _{k=1}^n\left(k\times \frac{\frac{n}{k}(\frac{n}{k}+1)}{2}\times \frac{\frac{m}{k}(\frac{m}{k}+1)}{2}\times \sum _{d|k}d\mu (d)\right)$$

后面的 ${\sum }_{d|k}d\mu (d)$ 可以用狄利克雷前缀和计算，也可以用积性函数计算

时间复杂度 $O(n)$

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code

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 10000100
#define mod 20101009
using namespace std;
ll n,m,w,ans,p[N],g[N],mu[N],prime[N];
const ll MX=1e7;
void work()
{mu[1]=1;for(ll i=2;i<=MX;++i){if(!p[i]){mu[i]=-1;prime[++w]=i;}for(ll j=1;j<=w&&i*prime[j]<=MX;++j){p[i*prime[j]]=1;if(i%prime[j]==0)break;else mu[i*prime[j]]=-mu[i];}}
//	for(ll i=1;i<=MX;++i)
//		for(ll j=1;i*j<=MX;++j)
//			g[i*j]+=i*mu[i];for(ll i=1;i<=MX;++i)g[i]=(i*mu[i]+mod)%mod;for(ll i=1;i<=w;++i)for(ll j=1;j*prime[i]<=MX;++j)(g[j*prime[i]]+=g[j])%=mod;return;
}
ll get(ll x)
{return (x*(x+1)/2)%mod;
}
int main()
{scanf("%lld%lld",&n,&m);work();for(ll i=1;i<=min(n,m);++i)(ans+=i*get(n/i)%mod*get(m/i)%mod*g[i]%mod)%=mod;printf("%lld",ans);return 0;
}