#欧拉函数#洛谷 2303 [SDOI2012] Longge 的问题

题目

求\(\sum_{i=1}^n\gcd(n,i)\)

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分析

\(=\sum_{i=1}^n\sum_{d|gcd(n,i)}\varphi(d)\)

\(=\sum_{d|n}\varphi(d)\sum_{i=1}^{\frac{n}{d}}1=\sum_{d|n}\varphi(d)\frac{n}{d}\)

这显然可以\(O(\sqrt{n}\log n)\)实现

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代码

#include <cstdio>
#include <cctype>
#include <map>
#define rr register
using namespace std;
typedef long long lll; map<lll,bool>uk;
lll n,nn,ans,Cnt,prime[31];
inline void dfs(lll rest,lll now,lll phi){
	if (uk[now]) return;
	ans+=phi*rest,uk[now]=1;
	if (now==n) return;
	for (rr int i=1;i<=Cnt;++i)
	if (rest%prime[i]==0){
		if (now%prime[i]==0) dfs(rest/prime[i],now*prime[i],phi*prime[i]);
		    else dfs(rest/prime[i],now*prime[i],phi*(prime[i]-1));
	}
}
signed main(){
	scanf("%lld",&n),nn=n;
	for (rr lll i=2;i*i<=nn;++i)
	if (nn%i==0){
		while (nn%i==0) nn/=i;
		prime[++Cnt]=i;
	}
	if (nn>1) prime[++Cnt]=nn;
	dfs(n,1,1);
	return !printf("%lld",ans);
}