gcd经典推导 hdu 5382

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多校训练 中有一道数论题，推导十分强大，特此记录。

题目链接 和推导过程:
proof
代码如下:

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//#pragma comment(linker,"/STACK:10240000,10240000")
#include <cstring>
#include <cstdio>
#include <cstdlib>
#include <ctime>
#include <cmath>
#include <iostream>
#include <algorithm>
#include <vector>
#include <map>
#include <set>
#include <queue>
#include <stack>
#include <string>
#include <bitset>
using namespace std;
typedef long long LL;
typedef pair<int,int> PII;
typedef pair<LL,LL> PLL;
#define clr(a,b) memset(a,b,sizeof(a))
#define MP make_pair
#define PB push_back
#define lrt rt<<1
#define rrt rt<<1|1
#define lson l,m,lrt
#define rson m+1,r,rrt
/*------ Welcome to visit blog of dna049: http://dna049.com ------*/
const int M = 258280327;
const int N = 1e6+6;
int a[N],g[N],t[N];
void init(){
    for(int i=0;i<N;++i){
        g[i]=1;t[i]=0;
    }
    for(int i=2;i<N;++i){
        if(g[i] == 1){
            for(int j=i;j<N;j+=i){
                g[j]<<=1;
            }
        }
    }
    for(int i=2;i<N;++i){
        for(int j=i;j<N;j+=i){
            t[j]=(t[j]+g[i-1])%M;
        }
    }
    a[1]=1;
    for(int i=2;i<N;++i){
        a[i]=(a[i-1]+2*i-1-t[i-1])%M;
        if(a[i]<0)  a[i]+=M;
    }
    for(int i=2;i<N;++i){
        a[i]=(a[i]+a[i-1])%M;
    }
}
int main(){
    //    freopen("/Users/dna049/Desktop/AC/in","r",stdin);
    //    freopen("/Users/dna049/Desktop/AC/out","w",stdout);
    init();
    int T;
    scanf("%d",&T);
    while(T--){
        int n;
        scanf("%d",&n);
        printf("%d\n",a[n]);
    }
    return 0;
}