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zc
2019-12-21 19:47:00
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GCD-Extreme top: 0

g(n) = \sum_{i=1}^{n-1} \gcd(i,n)

$$

\sum{i=1}^{n-1} \sum{j=i+1}^n \gcd(i,j)

\\

= \sum_{i=1}^n g(i)

$$

$$

g(n) = \sum_{i=1}^{n-1} \gcd(i,n)

\\

= \sum{d|n}d \sum{i=1}^{n-1}[gcd(i,n)=d]

\\

= \sum{d|n}d \sum{i=1}^{\frac nd - 1}[gcd(i,n)=1]

\\

= \sum_{d|n}d \times \varphi(\frac nd);

$$

求出g的前缀和

可以用筛的方式求g

O(n \sqrt n + T);

不能用之前的套路,否则会tle

#include<bits/stdc++.h>
namespace ZDY{
    #pragma GCC optimize(3)
    #define il __inline__ __attribute__ ((always_inline))
    #define rg register
    #define ll long long
    #define ull unsigned long long
    #define db double
    #define sht short
    #define MB template <class T>il
    #define Fur(i,x,y) for(int i=x;i<=y;i++)
    #define Fdr(i,x,y) for(int i=x;i>=y;i--)
    #define fl(i,x) for(int i=head[x],to;to=e[i].to,i;i=e[i].nxt)
    #define clr(x,y) memset(x,y,sizeof(x))
    #define cpy(x,y) memcpy(x,y,sizeof(x))
    #define fin(s) freopen(s".in","r",stdin)
    #define fout(s) freopen(s".out","w",stdout)
    #define fcin ios::sync_with_stdio(false)
    #define l2(n) (int(log2(n)))
    #define inf 0x3f3f3f3f
    MB T ABS(T x){return x>0?x:-x;}
    MB T MAX(T x,T y){return x>y?x:y;}
    MB T MIN(T x,T y){return x<y?x:y;}
    MB T GCD(T x,T y){return y?GCD(y,x%y):x;}
    MB void SWAP(T &x,T &y){T t=x;x=y;y=t;}
}using namespace ZDY;using namespace std;
namespace IO{const char* ln="\n";const int str=1<<20;struct IN{char buf[str],*s,*t;bool _;IN():s(buf),t(buf),_(0){}il char gc(){return s==t&&((t=(s=buf)+fread(buf,1,str,stdin))==s)?EOF:(*s++);}IN&operator>>(char&ch){if(_)return *this;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF)_=1;else ch=c;return *this;}IN& operator>>(char* ch){if(_)return *this;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF)return _=1,*this;*ch=c;ch++;while((c=gc())!=EOF&&!isspace(c))*ch=c,ch++;if(c==EOF)_=1;return *this;}IN& operator>>(string& ch){if(_)return *this;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF)return _=1,*this;ch+=c;while((c=gc())!=EOF&&!isspace(c))ch+=c;if(c==EOF)_=1;return *this;}template<typename T>IN&operator>>(T&x){if(_)return *this;char c=gc();bool ff=0;while(c!=EOF&&(c<'0'||c>'9'))ff^=(c=='-'),c=gc();if(c==EOF){_=1;return *this;}x=0;while(c!=EOF&&'0'<=c&&c<='9')x=(x<<3)+(x<<1)+c-48,c=gc();if(c==EOF)_=1;if(ff)x=-x;return *this;}}in;struct OUT{char buf[str],*s,*t;OUT():s(buf),t(buf+str){}~OUT(){fwrite(buf,1,s-buf,stdout);}void pt(char c){(s==t)?(fwrite(s=buf,1,str,stdout),*s++=c):(*s++=c);}OUT&operator<<(const char*s){while(*s)pt(*s++);return *this;}OUT&operator<<(char*s){while(*s)pt(*s++);return *this;}OUT&operator<<(string s){for(int i=0;s[i];i++)pt(s[i]);return *this;}template<typename T>OUT&operator<<(T x){if(!x)return pt('0'),*this;if(x<0)pt('-'),x=-x;char a[30],t=0;while(x)a[t++]=x%10,x/=10;while(t--)pt(a[t]+'0');return *this;}}out;}using namespace IO;
#define N 1000011
int phi[N],pri[N],tot=0;
ll ans[N];
bool f[N];
void ol(int n){
    phi[1]=0;
    Fur(i,2,n){
        if(!f[i])phi[i]=i-1,pri[++tot]=i;
        Fur(j,1,tot){
            if(pri[j]*i>n)break;
            f[pri[j]*i]=1;
            if(i%pri[j])phi[i*pri[j]]=phi[i]*phi[pri[j]];
            else{
                phi[i*pri[j]]=phi[i]*pri[j];
                break;
            }
        }
    }
    Fur(i,1,n)ans[i]=phi[i];
    int len=sqrt(n);
    Fur(i,2,len){
        ans[i*i]+=phi[i]*i;
        for(int j=i+1;i*j<=n;j++)
        ans[i*j]+=phi[i]*j+phi[j]*i;
    }
    ans[1]=0;
    Fur(i,2,n)ans[i]+=ans[i-1];
}
int main(){
    ol(1000000);
    int n;in>>n;
    while(n){
        out<<ans[n]<<ln;
        in>>n;
    }
}
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