SP34096 【DIVCNTK - Counting Divisors (general)】

题目

求

\[\sum_{i=1}^n \sigma(i^k) \]

我们先来设一个函数\(f(i)=\sigma(i^k)\)

根据约数个数定理

\[f(p)=\sigma(p^k)=k+1 \]

\[f(p^c)=\sigma(p^{ck})=ck+1 \]

这不就可以Min_25筛了吗

还是先求出来一个区间内的质数个数，一个质数的贡献显然是\(k+1\)，之后上板子就好了

代码

#include<algorithm>
#include<iostream>
#include<cstring>
#include<cstdio>
#include<cmath>
#define maxn 3000005
#define re register
#define LL unsigned long long
int T,tot,f[maxn];
LL n,K,p[maxn],w[maxn],id1[maxn],id2[maxn],m,Sqr,g[maxn];
inline LL S(LL x,int y) {
    if(x<=1||p[y]>x) return 0;
    int t=(x<=Sqr)?id1[x]:id2[n/x];
    LL ans=g[t]*(K+1)-(K+1)*(y-1);
    for(re int k=y;k<=tot&&p[k]*p[k]<=x;k++) {
        LL p1=p[k];
        for(re int e=1;p1<=x;e++,p1*=p[k])
            ans+=(S(x/p1,k+1)+((e>1)?1:0))*(K*e+1);
    }
    return ans;
}
int main()
{
    scanf("%d",&T);f[1]=1;
    for(re int i=2;i<=100000;i++) {
        if(!f[i]) p[++tot]=i;
        for(re int j=1;j<=tot&&p[j]*i<=100000;j++) {
            f[p[j]*i]=1;if(i%p[j]==0) break;
        }
    }
    while(T--) {
        scanf("%llu%llu",&n,&K);Sqr=std::sqrt(n)+1;m=0;
        for(re LL l=1,r;l<=n;l=r+1) {
            r=n/(n/l);w[++m]=n/l;
            if(w[m]<=Sqr) id1[w[m]]=m;
                else id2[n/w[m]]=m;
            g[m]=w[m]-1;
        }
        for(re int j=1;j<=tot&&p[j]*p[j]<=n;j++) 
            for(re int i=1;i<=m&&p[j]*p[j]<=w[i];i++) {
                int k=(w[i]/p[j]<=Sqr)?id1[w[i]/p[j]]:id2[n/(w[i]/p[j])];
                g[i]=g[i]-g[k]+j-1;
            }
        printf("%llu\n",S(n,1)+1);
    }
    return 0;
}