正题

P1891

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题目大意

有T个询问，每个询问给出n，求$$\sum _{i=1}^{n}lcm(i,n)$$

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解题思路

$$\sum _{i=1}^{n}lcm(i,n)$$
$$n\sum _{i=1}^{n}i/gcd(i,n)$$
$$n\sum _{d|n}\frac{1}{d}\sum _{i=1}^{n}i[gcd(i,n)=d]$$
对互质的数求和
$$n\sum _{d|n}\frac{1}{d}\times \frac{\phi (\frac{n}{d})n}{2}$$
$$n\sum _{d|n}\frac{\phi (d)\times d}{2}$$

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code

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define ll long long
#define N 1000100
using namespace std;
ll T,n,w,p[N],prime[N],phi[N],g[N];
const ll MX=1e6;
void work()
{phi[1]=1;for(ll i=2;i<=MX;++i){if(!p[i]){prime[++w]=i;phi[i]=i-1;}for(ll j=1;j<=w&&i*prime[j]<=MX;++j){p[i*prime[j]]=1;if(i%prime[j]==0){phi[i*prime[j]]=phi[i]*prime[j];break;}else phi[i*prime[j]]=phi[i]*(prime[j]-1);}}for(ll i=1;i<=MX;++i)for(ll j=1;i*j<=MX;++j)g[i*j]+=(i==1?1:i*phi[i]/2);//1只有一个数return;
}
int main()
{work();scanf("%lld",&T);while(T--){scanf("%lld",&n);printf("%lld\n",n*g[n]);}return 0;
}