bzoj3944 Sum

http://www.elijahqi.win/archives/3164
题目描述

给定一个正整数

N(N\le2^{31}-1)

N(N≤231−1)

求

ans_1=\sum_{i=1}^n\phi(i),ans_2=\sum_{i=1}^n \mu(i)

ans1​=∑i=1n​ϕ(i),ans2​=∑i=1n​μ(i)

输入输出格式

输入格式：

一共T+1行 第1行为数据组数T(T<=10) 第2~T+1行每行一个非负整数N，代表一组询问

输出格式：

一共T行，每行两个用空格分隔的数ans1,ans2

输入输出样例

输入样例#1： 复制

6
1
2
8
13
30
2333
输出样例#1： 复制

1 1
2 0
22 -2
58 -3
278 -3
1655470 2
mmp 最后一个点是inf 然后爆int了
求$\varphi$的公式在这里
https://blog.youkuaiyun.com/elijahqi/article/details/79610235
求$\mu$同理
$\sum\limits_{i=1}^{n}\sum\limits_{d|n}\mu(\frac{n}{d})\times 1(d)=\sum\limits_{i=1}^{n}\varepsilon (i)$
枚举d
$\sum\limits_{d=1}^{n}\sum\limits_{i=1}^{\frac{n}{d}}\mu(i)=\sum\limits_{i=1}^{n}\varepsilon (i)$
设Hi表示$\mu$的前缀
即$\sum\limits_{d=1}^{n}H(\frac{n}{d})=1$

#include<cstdio>
#include<cctype>
#include<cstring>
#include<algorithm>
#define ll long long
using namespace std;
inline char gc(){
    static char now[1<<16],*S,*T;
    if (T==S){T=(S=now)+fread(now,1,1<<16,stdin);if (T==S) return EOF;}
    return *S++;
}
inline int read(){
    int x=0,f=1;char ch=gc();
    while(!isdigit(ch)) {if (ch=='-') f=-1;ch=gc();}
    while(isdigit(ch)) x=x*10+ch-'0',ch=gc();
    return x*f;
}
int T;
const int N=2e6+10;
const int M=100011;
int prime[N],tot,n;bool not_prime[N];
ll phi[N],s[N],s1[N],mu[N];bool visit[N];
inline ll getp(ll x){
    if (x<=2e6) return phi[x];else return s[n/x];
}
inline ll getm(ll x){
    if (x<=2e6) return mu[x];else return s1[n/x];
}
inline void calc(int x){
    if (x<=2e6) return;if(visit[n/x]) return;
    visit[n/x]=1;ll tmp=(ll)x*x+x>>1,tmp1=1;unsigned int last;
    for (unsigned int i=2;i<=x;i=last+1){
        last=x/(x/i);calc(x/i);
        tmp-=getp(x/i)*(last-i+1);
        tmp1-=getm(x/i)*(last-i+1);
    }s[n/x]=tmp;s1[n/x]=tmp1;
}
const int out_len=1<<16;
char obuf[out_len],*oh=obuf;
inline void write_char(char c){
    if (oh==obuf+out_len) fwrite(obuf,1,out_len,stdout),oh=obuf;
    *oh++=c;
}
template<class T>
inline void W(T x){
    static int buf[30],cnt;
    if (!x) write_char('0');else{
        if (x<0) write_char('-'),x=-x;
        for (cnt=0;x;x/=10) buf[++cnt]=x%10+48;
        while(cnt) write_char(buf[cnt--]);
    }
}
inline void flush(){fwrite(obuf,1,oh-obuf,stdout);}
int main(){
    freopen("bzoj3944.in","r",stdin);
//  freopen("bzoj3944.out","w",stdout);
    T=read();phi[1]=mu[1]=1;
    for (int i=2;i<=2e6;++i){
        if (!not_prime[i]) prime[++tot]=i,phi[i]=i-1,mu[i]=-1;
        for (int j=1;prime[j]*i<=2e6;++j){
            not_prime[prime[j]*i]=1;
            if (i%prime[j]==0) {mu[prime[j]*i]=0,phi[prime[j]*i]=phi[i]*prime[j];break;}
            else mu[prime[j]*i]=-mu[i],phi[prime[j]*i]=phi[i]*phi[prime[j]];
        }
    }for (int i=2;i<=2e6;++i) phi[i]+=phi[i-1],mu[i]+=mu[i-1];
   // for (int i=2;i<=2e6;++i) printf("%d\n",mu[i]);
    while(T--){
        memset(visit,0,sizeof(visit));
        n=read();if (n<=2e6) W(phi[n]),write_char(' '),W(mu[n]),write_char('\n');
        else calc(n),W(s[1]),write_char(' '),W(s1[1]),write_char('\n');
        //printf("%lld %lld\n",s[1],s1[1]);
        //printf("%lld %lld\n",calcphi(n),calcmu(n));
    }flush();
    return 0;
}