P2303 [SDOI2012]Longge的问题 我傻QwQ

莫比乌斯反演学傻了$QwQ$

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思路：推式子？

提交：2次

错因：又双叒叕没开$long\space long$

题解：

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

$=\sum_{d|n}d\sum_{i=1}^{\frac{n}{d}} [gcd(i,\frac{n}{d})=1]$

注意到$\sum_{i=1}^{\frac{n}{d}} [gcd(i,\frac{n}{d})=1]$就是与$\frac{n}{d}$互质的数的个数。

$=\sum_{d|n}d\varphi(\frac{n}{d})$

代码：

#include<cstdio>
#include<iostream>
#include<cmath>
using namespace std;
#define ull unsigned long long
#define ll long long
#define R register ll
#define pause (for(R i=1;i<=10000000000;++i))
#define In freopen("NOIPAK++.in","r",stdin)
#define Out freopen("out.out","w",stdout)
namespace Fread {
static char B[<<],*S=B,*D=B;
#ifndef JACK
#define getchar() (S==D&&(D=(S=B)+fread(B,1,1<<15,stdin),S==D)?EOF:*S++)
#endif
inline ll g() {
    R ret=,fix=; register char ch; while(!isdigit(ch=getchar())) fix=ch=='-'?-:fix;
    if(ch==EOF) return EOF; do ret=ret*+(ch^); while(isdigit(ch=getchar())); return ret*fix;
} inline bool isempty(const char& ch) {return (ch<=||ch>=);}
inline void gs(char* s) {
    register char ch; while(isempty(ch=getchar()));
    do *s++=ch; while(!isempty(ch=getchar()));
}
} using Fread::g; using Fread::gs;
namespace Luitaryi {
const int N=<<+;
int n,cnt; ll ans;
int pri[N>>]; bool vis[N];
inline void PRE(int n) {
    for(R i=;i<=n;++i) {
        if(!vis[i]) pri[++cnt]=i;
        for(R j=;j<=cnt&&i*pri[j]<=n;++j) {
            vis[i*pri[j]]=true; if(i%pri[j]==) break;
        }
    }
}
inline ll phi(ll x) { R ret=x;
    for(R i=;i<=cnt&&1ll*pri[i]*pri[i]<=x;++i) if(x%pri[i]==) {
        ret=ret/pri[i]*(pri[i]-); while(x%pri[i]==) x/=pri[i];
    } if(x>) ret=ret/x*(x-); return ret;
}
inline void main() {
    n=g(); R lim=sqrt(n); PRE(lim);
    for(R i=;i<=lim;++i) if(n%i==) {
        ans+=i*phi(n/i);
        if(i*i!=n) ans+=n/i*phi(i);
    } printf("%lld\n",ans);
}
}
signed main() {
    Luitaryi::main();
}

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2019.07.18