Problem 521 | Project Euler |

[sx349]

https://pe-cn.github.io/521/

Problem 521 Smallest prime factor Let smpf(n) be the smallest prime factor of n.smpf(91)=7 because 91=7×13 and smpf(45)=3 because 45=3×3×5.Let S(n) be the sum of smpf(i) for 2 ≤ i ≤ n.E.g. S(100)=125

[shangkelingxiang]

#include<bits/stdc++.h>
using namespace std;
#define debug(x) cerr<<#x<<' '<<x<<endl
#ifndef ONLINE_JUDGE
    #define pia getchar
#else
    #define pia nc
#endif
char nc(){
    static char buf[1<<25],*p=buf,*q=buf;
    if(p==q&&(q=(p=buf)+fread(buf,1,1<<25,stdin),p==q))return EOF;
    return *p++;
}
template<class T>void rd(T&x){
    short f=1;x=0;
    char ch=pia();
    while(!isdigit(ch)){
        if(ch=='-')f=-1;
        ch=pia();
    }while(isdigit(ch)){
        x=(x<<1)+(x<<3)+(ch^48);
        ch=pia();
    }x*=f;
}
template<class T>void wr(T x){
    if(x<0)putchar('-'),x=-x;
    if(x>=10)wr(x/10);
    putchar(x%10+48);
}
#define int __int128
#define maxn 2333333
const int p=1e9;
int lpf[maxn],pri[maxn],pcnt,spri[maxn],mx;
void shai(int n){
    for(int i=2;i<=n;i++){
        if(!lpf[i])pri[lpf[i]=++pcnt]=i,spri[pcnt]=spri[pcnt-1]+i;
        for(int j=1;j<=lpf[i]&&i*pri[j]<=n;j++)lpf[i*pri[j]]=j;
    }
}
__int64 N,lim;
int ans;
int ge[maxn],le[maxn],tot;
#define id(x) ((x)<=lim?le[x]:ge[N/(x)])
int li[maxn],G[maxn][2];
void init(int n){
    tot=0;
    for(int i=1,j;i<=n;i=n/j+1){
        j=n/i;
        id(j)=++tot;
        li[tot]=j;
        G[tot][0]=j%p-1;
        if(j&1)G[tot][1]=(j-1)/2*(j+2)%p;
        else G[tot][1]=(j+2)/2*(j-1)%p;
    }
}
void calcFprime(){
    for(int k=1;pri[k]<=lim;k++)
    for(int i=1;li[i]>=pri[k]*pri[k];i++){
        int d=id(li[i]/pri[k]);
        if(i==1)ans=(ans+pri[k]*(G[d][0]-(k-1)+p))%p;
        G[i][0]-=G[d][0]-(k-1)-p;
        G[i][1]-=pri[k]*(G[d][1]-spri[k-1])%p-p;
        G[i][0]%=p,G[i][1]%=p;
    }
}
signed main(){
  cin>>N;
    lim=sqrt(N);
    shai(lim+1000);
    init(N);
    ans=0;
    calcFprime();
    ans+=G[1][1];
    wr(ans%p);
}

[shangkelingxiang]

min25筛即可，原题是https://www.luogu.com.cn/problem/SP19975