BZOJ 3944 Sum

杜教筛第一道题！

题面：

3944: Sum

Time Limit: 10 Sec  Memory Limit: 128 MB
Submit: 3843  Solved: 1030
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Description

Input

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

Output

一共T行，每行两个用空格分隔的数ans1,ans2

Sample Input

6
1
2
8
13
30
2333

Sample Output

1 1
2 0
22 -2
58 -3
278 -3
1655470 2

HINT

反演+杜教筛。

$1=sum_{i=1}^{n}sum_{d|i}mu(d)=sum_{d=1}^{n}sum_{i=1}^{lfloorfrac{n}{d} floor}mu(i)=sum_{i=1}^{n}S(lfloorfrac{n}{i} floor)$

$ herefore S(n)=1-sum_{i=2}^{n}S(lfloorfrac{n}{i} floor)$

$S(n)=sum_{i=1}^{n}varphi(i)=sum_{i=1}^{n}[i-sum_{d|i,d<i}varphi(d)]=frac{n(n+1)}{2}-sum_{i=2}^{n}sum_{d|i}varphi(d)=frac{n(n+1)}{2}-sum_{i=2}^{n}S(lfloorfrac{n}{i} floor)$

之后预处理前$frac{2}{3}$，分块回答询问(记忆化搜索)。

#include<iostream>
#include<stdio.h>
#include<string.h>
using namespace std;
#define LL long long
#define maxn 5400000
#define maxm 100000
#define uint unsigned int
int T,n;
int p[670000],cnt;
LL mu[maxn+1],varphi[maxn+1];
LL ans_mu[maxm+1],ans_varphi[maxm+1];
bool book[maxn+1],quest_mu[maxm+1],quest_varphi[maxm+1];
void init()
{
    mu[1]=1;
    varphi[1]=1;
    for(int i=2;i<=maxn;i++)
    {
        if(!book[i])
        {
            mu[i]=-1;
            varphi[i]=i-1;
            p[++cnt]=i;
        }
        for(int j=1;j<=cnt&&1LL*p[j]*i<=maxn;j++)
        {
            book[i*p[j]]=true;
            if(i%p[j])
            {
                mu[i*p[j]]=-mu[i];
                varphi[i*p[j]]=varphi[i]*(p[j]-1);
            }   
            else
            {
                mu[i*p[j]]=0;
                varphi[i*p[j]]=varphi[i]*p[j];
                break;
            }
        }
    }
    for(int i=2;i<=maxn;i++)
        mu[i]+=mu[i-1],varphi[i]+=varphi[i-1];
}
LL questmu(uint now)
{
    if(now<=maxn)
        return mu[now];
    int last=n/now,nex;
    if(quest_mu[last])
        return ans_mu[last];
    LL save=1;
    for(uint i=2;i<=now;i=nex+1)
    {
        nex=now/(now/i);
        save-=questmu(now/i)*(nex-i+1);
    }   
    quest_mu[last]=true;
    ans_mu[last]=save;
    return save;
}
LL questvarphi(uint now)
{
    if(now<=maxn)
        return varphi[now];
    int last=n/now,nex;
    if(quest_varphi[last])
        return ans_varphi[last];
    LL save=(LL)now*(now+1)>>1;
    for(int i=2;i<=now;i=nex+1)
    {
        nex=now/(now/i);
        save-=questvarphi(now/i)*(nex-i+1);
    }
    quest_varphi[last]=true;
    ans_varphi[last]=save;
    return save;
}
int main()
{
    init();
    scanf("%d",&T);
    while(T--)
    {
        scanf("%d",&n);
        memset(quest_mu,false,sizeof(quest_mu));
        memset(quest_varphi,false,sizeof(quest_varphi));
        printf("%lld %lld
",questvarphi(n),questmu(n));
    }
}