正题

题目链接:
https://www.lydsy.com/JudgeOnline/problem.php?id=2226
https://www.luogu.org/problem/SP5971

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题目大意

求$$\sum _{i=1}^{n}lcm(n,i)$$

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解题思路

$$\sum _{i=1}^{n}lcm(n,i)$$
$$\sum _{i=1}^n\frac{ni}{gcd(n,i)}$$
$$n\sum _{d\mid n}\sum _{i=1}^n[(n,i)==d]i$$
$$n\sum _{d\mid n}\sum _{i=1}^n[(n/d,i/d)==1]i$$
$$n\sum _{d|n}\frac{d\ast \phi (d)}{2}$$

然后预处理$O(n\log n)$然后$O(1)$回答即可

时间复杂度$O(n\log n+T)$

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$code$

#include<cstdio>
#include<cstring>
#include<algorithm>
#define ll long long
using namespace std;
const ll N=1e6; 
ll T,phi[N+10],ans[N+10],n;
int main()
{for (ll i=2;i<=N;i++) phi[i]=i;for (ll i=2;i<=N;i++){bool flag=(phi[i]==i);for (ll j=i;j<=N;j+=i){if(flag)phi[j]=phi[j]/i*(i-1);ans[j]+=phi[i]*i/2;}}scanf("%lld",&T);while(T--){scanf("%lld",&n);printf("%lld\n",n*(ans[n]+1));}
}