4805: 欧拉函数求和
Time Limit: 15 Sec Memory Limit: 256 MB
Submit: 614 Solved: 342
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Description
给出一个数字N，求sigma(phi(i)),1<=i<=N
Input
正整数N。N<=2*10^9
Output
输出答案。
Sample Input
10
Sample Output
32
HINT
Source
直接大力杜教筛
$\sum_{i=1}^{n}\varphi(i) = \frac{n\times(n+1)}{2} - \sum_{d=2}^{n}\sum_{i=1}^{\lfloor\frac{n}{d}\rfloor}\varphi(i)$
#include<cstdio>
#include<map>
#include<ext/pb_ds/assoc_container.hpp>
#include<ext/pb_ds/hash_policy.hpp>
#define LL long long
using namespace std;
using namespace __gnu_pbds;
const int MAXN=;
int N,limit=,tot=,vis[MAXN],prime[MAXN];
LL phi[MAXN];
gp_hash_table<int,LL>Aphi;
void GetPhi()
{
vis[]=;phi[]=;
for(int i=;i<=limit;i++)
{
if(!vis[i]) prime[++tot]=i,phi[i]=i-;
for(int j=;j<=tot&&i*prime[j]<=limit;j++)
{
vis[i*prime[j]]=;
if(i%prime[j]==) {phi[i*prime[j]]=phi[i]*prime[j];break;}
else phi[i*prime[j]]=phi[i]*(prime[j]-);
}
}
for(int i=;i<=limit;i++) phi[i]+=phi[i-];
}
LL SolvePhi(LL n)
{
if(n<=limit) return phi[n];
if(Aphi[n]) return Aphi[n];
LL tmp=n*(n+)/;
for(int i=,nxt;i<=n;i=nxt+)
{
nxt=min(n,n/(n/i));
tmp-=SolvePhi(n/i)*(LL)(nxt-i+);
}
return Aphi[n]=tmp;
}
int main()
{
GetPhi();
scanf("%lld",&N);
printf("%lld",SolvePhi(N));
return ;
}