[BZOJ2705][SDOI2012]Longge的问题（数论）

题目描述

传送门

题解

基础的数论题。
∑i=1n(i,n)
=∑i=1n∑d=1n[(i,n)=d]d
=∑d=1n∑i=1n[(id,nd)=1]d
令i=di
=∑d=1n∑i=1nd[(i,nd)=1]d
=∑d=1nϕ(nd)d
暴力枚举约数，根n求phi

代码

#include<iostream>#include<cstring>#include<cstdio>using namespace std;#define LL long longLL n,ans;inline LL phi(LL x){    LL ans=x;    for (LL i=2;i*i<=n;++i)      if (x%i==0){        ans=ans*(i-1)/i;        while (x%i==0) x/=i;      }    if (x>1) ans=ans*(x-1)/x;    return ans;}int main(){    scanf("%I64d",&n);    for (LL i=1;i*i<=n;++i)      if (n%i==0){        ans+=phi(n/i)*i;        if (i!=n/i) ans+=phi(i)*(n/i);      }    printf("%I64d\n",ans);}