Battlestation Operational

Battlestation Operational

Time Limit: 6000/3000 MS (Java/Others)    Memory Limit: 65536/65536 K (Java/Others)

Problem Description

> The Death Star, known officially as the DS-1
Orbital Battle Station, also known as the Death Star I, the First Death Star,
Project Stardust internally, and simply the Ultimate Weapon in early development
stages, was a moon-sized, deep-space mobile battle station constructed by the
Galactic Empire. Designed to fire a single planet-destroying superlaser powered
by massive kyber crystals, it was the pet project of the Emperor, Darth Vader,
and its eventual commander Grand Moff Wilhuff Tarkin to expound the military
philosophy of the aptly named Tarkin Doctrine.
>
> —
Wookieepedia

In the story of the Rogue One, the rebels risked their lives
stolen the construction plan of the Death Star before it can cause catastrophic
damage to the rebel base. According to the documents, the main weapon of the
Death Star, the Superlaser, emits asymmetric energy in the battlefield that
cause photons to annihilate and burns everything in a single shot.

You
are assigned the task to estimate the damage of one shot of the Superlaser.

Assuming that the battlefield is an n×n

grid. The energy field ignited by the Superlaser is asymmetric over the grid.
For the cell at i

-th row and j

-th column, ⌈i/j⌉

units of damage will be caused. Furthermore, due to the quantum effects, the
energies in a cell cancel out if gcd(i,j)≠1

or i<j

.

The figure below illustrates the damage caused to each cell for n=100

. A cell in black indicates that this cell will not be damaged due to the
quantum effects. Otherwise, different colors denote different units of
damages.

Your should calculate
the total damage to the battlefield. Formally, you should compute

f(n)=∑i=1n∑j=1i⌈ij⌉[(i,j)=1],

where [(i,j)=1]

evaluates to be 1

if gcd(i,j)=1

, otherwise 0

.

 

Input

There are multiple test cases.

Each line of the
input, there is an integer n

(1≤n≤106

), as described in the problem.

There are up to 104

test cases.

 

Output

For each test case, output one integer in one line
denoting the total damage of the Superlaser, f(n) mod 109+7

.

 

Sample Input

1
2
3
10

 

Sample Output

1
3
8
110

分析：莫比乌斯反演，d(n)=Σ(d|n) f(d) ；

　　　d(n)=Σi = (1~n)Σj = (1~i) i/j , f(n) = Σi = (1~n)Σj = (1~i) [gcd(i,j)==1] i/j； 

　　　要求 f(n), f(n) = d(n) - Σ(d|n) f(d) ( d != n)；

代码：

#include<cstdio>
#define mod 1000000007
#define rep(i,m,n) for(i=m;i<=(int)n;i++)
const int maxn=1e6+;
int n,m,k,t;
long long p[maxn];
void init()
{
    int i,j;
    rep(i,,maxn-)
    {
        p[i]++;
        p[i+]--;
        for(int j=i,cnt=;j+<=maxn-;j+=i,cnt++)
        {
            p[j+]+=cnt;
            if(j+i+<=maxn-)p[j+i+]-=cnt;
        }
    }
    rep(i,,maxn-)p[i]=(p[i]%mod+p[i-]+mod)%mod;
    rep(i,,maxn-)
    {
        for(j=i*;j<=maxn-;j+=i)
        {
            p[j]=(p[j]-p[i]+mod)%mod;
        }
    }
    rep(i,,maxn-)(p[i]+=p[i-])%=mod;
}
int main()
{
    int i,j;
    init();
    while(~scanf("%d",&n))printf("%lld\n",p[n]);
    return ;
}