BZOJ2226:LCMSum(欧拉函数)

Description

Given n, calculate the sum LCM(1,n) + LCM(2,n) + .. + LCM(n,n), where LCM(i,n) denotes the Least Common Multiple of the integers i and n.

Input

The first line contains T the number of test cases. Each of the next T lines contain an integer n.

Output

Output T lines, one for each test case, containing the required sum.

Sample Input

3
1
2
5

Sample Output

1
4
55

HINT

Constraints
1 <= T <= 300000
1 <= n <= 1000000

Solution

$sum_{i=1}^{n}lcm(i,n)$
$=sum_{i=1}^{n}frac{i imes n}{gcd(i,n)}$
$=frac{1}{2}(sum_{i=1}^{n-1}frac{i imes n}{gcd(i,n)}+sum_{i=n-1}^{1}frac{i imes n}{gcd(i,n)})+n$
因为$gcd(a,b)=gcd(a-b,b)$，所以上面的两个$sum$可以合起来。
$=frac{1}{2}sum_{i=1}^{n-1}frac{n^2}{gcd(i,n)}+n$
设$gcd(i,n)=d$，把式子改为枚举$d$,那么与$n$的$gcd$为$d$的数有$φ(frac{n}{d})$个。
$=frac{1}{2}sum_{d|n}frac{n^2 imes φ(frac{n}{d})}{d}+n$
设$d'=frac{n}{d}$，上下约分一下
$=frac{1}{2}sum_{d'|n}d' imes φ(d')+n$
预处理出$φ$数组，然后枚举每一个约数去计算它对它所有倍数的贡献，复杂度是调和级数的$O(nlogn)$。

Code

#include<iostream>
#include<cstring>
#include<cstdio>
#define N (1000009)
#define MAX (1000000)
#define LL long long
using namespace std;

inline int read()
{
    int x=0,w=1; char c=getchar();
    while (c<'0' || c>'9') {if (c=='-') w=-1; c=getchar();}
    while (c>='0' && c<='9') x=x*10+c-'0', c=getchar();
    return x*w;
}

LL T,n,cnt,phi[N],ans[N],vis[N],prime[N];

void Preprocess()
{
    phi[1]=1;
    for (int i=2; i<=MAX; ++i)
    {
        if (!vis[i]) prime[++cnt]=i, phi[i]=i-1;
        for (int j=1; j<=cnt && i*prime[j]<=MAX; ++j)
        {
            vis[i*prime[j]]=1;
            if (i%prime[j]) phi[i*prime[j]]=phi[i]*(prime[j]-1);
            else {phi[i*prime[j]]=phi[i]*prime[j]; break;}
        }
    }
    for (int i=1; i<=MAX; ++i)
        for (int j=i; j<=MAX; j+=i)
            ans[j]+=i*phi[i]/2;
    for (int i=1; i<=MAX; ++i) ans[i]=ans[i]*i+i;
}

int main()
{
    Preprocess();
    T=read();
    while (T--) n=read(), printf("%lld
",ans[n]);
}