Luogu P1447 [NOI2010]能量采集 数论？？欧拉

刚学的欧拉反演（在最后）就用上了，挺好$qwq$

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题意:求$\sum_{i=1}^{N}\sum_{j=1}^{M}(2*gcd(i,j)-1)$

原式

$=2*\sum_{i=1}^{N}\sum_{j=1}^{M}gcd(i,j)\space-m*n$

$=2*\sum_{i=1}^{N}\sum_{j=1}^M\sum_{d|gcd(i,j)}\varphi(d)\space-m*n$

$=2*\sum_{i=1}^{\lfloor \frac{N}{d}\rfloor}\sum_{j=1}^{\lfloor \frac{M}{d} \rfloor}\sum_{d=1}^N\varphi(d)\space-m*n$

$=2*\sum_{d=1}^N\varphi(d)\lfloor \frac{N}{d}\rfloor \lfloor \frac{M}{d} \rfloor \space-m*n$

所以又可以整除分块+线性筛$\varphi(n)$前缀和$

#include<cstdio>

#include<iostream>

#include<algorithm>

#include<cstring>

#include<cmath>

#include<cctype>

#include<cstdlib>

#include<vector>

#include<queue>

#include<map>

#include<set>

#define ll long long

#define R register int

using namespace std;

namespace Fread {

    static char B[<<],*S=B,*D=B;

    #define getchar() (S==D&&(D=(S=B)+fread(B,1,1<<15,stdin),S==D)?EOF:*S++)

    inline int g() {

        R ret=,fix=; register char ch; while(!isdigit(ch=getchar())) fix=ch=='-'?-:fix;

        do ret=ret*+(ch^); while(isdigit(ch=getchar())); return ret*fix;

    }

}using Fread::g;

const int N=;

ll p[N],pri[N],cnt;

bool v[N];

inline void PHI(int n) { p[]=;

    for(R i=;i<=n;++i) {

        if(!v[i]) pri[++cnt]=i,p[i]=i-;

        for(R j=;j<=cnt&&i*pri[j]<=n;++j) {

            v[i*pri[j]]=true;

            if(i%pri[j]==) {

                p[i*pri[j]]=pri[j]*p[i];

                break;

            } p[i*pri[j]]=p[i]*p[pri[j]];

        }

    } for(R i=;i<=n;++i) p[i]+=p[i-];

} int n,m;

ll ans;

signed main() {

#ifdef JACK

    freopen("NOIPAK++.in","r",stdin);

#endif

    PHI(); n=g(),m=g(); n>m?swap(n,m):void();

    for(R l=,r;l<=n;l=r+) {

        r=min(n/(n/l),m/(m/l));

        ans+=(ll)*(p[r]-p[l-])*(n/l)*(m/l);

    } printf("%lld\n",ans-(ll)n*m);

}

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2019.06.09