洛谷P1390_公约数的和

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有感觉了

两种不同公式的代码：

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#include <bits/stdc++.h>
/*
#include<ext/pb_ds/assoc_container.hpp>
#include<ext/pb_ds/hash_policy.hpp>
*/
using namespace std;

const double eps = 1e-10;
const double pi = 3.1415926535897932384626433832795;
const double eln = 2.718281828459045235360287471352;

#define f(i, a, b) for (int i = a; i <= b; i++)
#define scan(x) scanf("%d", &x)
#define mp make_pair
#define pb push_back
#define lowbit(x) (x&(-x))

#define fi first
#define se second
#define SZ(x) int((x).size())
#define all(x) x.begin(), x.end()
#define rall(x) x.rbegin(), x.rend()
#define summ(a) (accumulate(all(a), 0ll))

typedef unsigned long long ull;
typedef pair<int,int> pii;
typedef vector<int> vi;

using ll=long long;

ll n=2e6,mu[(int)2e6+9],pr[(int)2e6+9],cnt,ans, s[(int)2e6+9];
bitset<(int)1e7+9>vis;

void init(int x){
	mu[1]=1;
	for(ll i=2;i<=x;++i){
		if(!vis[i])pr[++cnt]=i,mu[i]=-1;
		for(int j=1;j<=cnt&&i*pr[j]<=x;++j){
			vis[pr[j]*i]=1;
			if(i%pr[j]==0)break;
			mu[i*pr[j]]=-mu[i];
		}
	}
	for(int i=1;i<=x;++i)for(ll j=1;j*i<=x;++j)s[j*i]+=1ll*mu[j]*i;
	for(int i=1;i<=x;++i)s[i]+=s[i-1];
}

int main()
{
    ios::sync_with_stdio(false);
    cin.tie(0);
    init(n);
    cin>>n;
    for(ll l=1,r;l<=n;l=r+1){
		r=n/(n/l);
		ans+=1ll*(n/l)*(n/l)*(s[r]-s[l-1]);
    }
    ans=(ans-1ll*(n+1ll)*n/2)/2;
    cout<<(ll)ans;
    return 0;
}
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#include <bits/stdc++.h>
/*
#include<ext/pb_ds/assoc_container.hpp>
#include<ext/pb_ds/hash_policy.hpp>
*/
using namespace std;

const double eps = 1e-10;
const double pi = 3.1415926535897932384626433832795;
const double eln = 2.718281828459045235360287471352;

#define f(i, a, b) for (int i = a; i <= b; i++)
#define scan(x) scanf("%d", &x)
#define mp make_pair
#define pb push_back
#define lowbit(x) (x&(-x))

#define fi first
#define se second
#define SZ(x) int((x).size())
#define all(x) x.begin(), x.end()
#define rall(x) x.rbegin(), x.rend()
#define summ(a) (accumulate(all(a), 0ll))

typedef unsigned long long ull;
typedef pair<int,int> pii;
typedef vector<int> vi;

using ll=long long;

ll n=2e6,mu[(int)2e6+9],s[(int)2e6+9],pr[(int)2e6+9],cnt,ans;
bitset<(int)1e7+9>vis;

void init(int x){
	mu[1]=1;
	for(ll i=2;i<=x;++i){
		if(!vis[i])pr[++cnt]=i,mu[i]=-1;
		for(int j=1;j<=cnt&&i*pr[j]<=x;++j){
			vis[pr[j]*i]=1;
			if(i%pr[j]==0)break;
			mu[i*pr[j]]=-mu[i];
		}
	}
	for(int i=1;i<=x;++i)s[i]=s[i-1]+mu[i];
}

ll cal(int x){
	ll res=0;
	for(ll l=1,k=n/x,r;l<=k;l=r+1){
		r=k/(k/l);
		res+=(k/l)*(k/l)*(s[r]-s[l-1]);
	}
	return res;
}

int main()
{
    ios::sync_with_stdio(false);
    cin.tie(0);
    init(n);
    cin>>n;
    for(int i=1;i<=n;++i)ans+=1ll*i*cal(i);
   	ans=(ans-1ll*n*(n+1ll)/2)/2;
    cout<<ans;
    return 0;
}