洛谷P1447 [NOI2010] 能量采集

洛谷P1447 [NOI2010] 能量采集

TITLE

思路

${\sum }_{i=1}^n{\sum }_{j=1}^{m}2\gcd (i,j)-1=-n\ast m+2{\sum }_{i=1}^n{\sum }_{j=1}^m\gcd (i,j)$

$\because {\sum }_{k|n}\varphi (k)=n$

$\therefore {\sum }_{i=1}^n{\sum }_{j=1}^m\gcd (i,j)={\sum }_{i=1}^n{\sum }_{j=1}^m{\sum }_{k|\gcd (i,j)}\varphi (k)={\sum }_{i=1}^n{\sum }_{j=1}^m{\sum }_{k|i,k|j}\varphi (k)={\sum }_{k=1}^{\min (n,m)}(n/i)\ast (m/j)\ast \varphi (k)$

$\therefore {\sum }_{i=1}^n{\sum }_{j=1}^{m}2\gcd (i,j)-1=-n\ast m+2{\sum }_{k=1}^{\min (n,m)}(n/i)\ast (m/j)\ast \varphi (k)$

$线性求\varphi$
$欧拉筛+\varphi 性质$
$\varphi 性质:$
(x is a prime)$\varphi (x)=x-1$
$(\gcd (x,y)=1)\varphi (x\ast y)=\varphi (x)\ast \varphi (y)$
(y is a prime,$y|x),\varphi (x\ast y)=\varphi (x)\ast y$

CODE

#include<iostream> 
#include<cstdio>
#include<cstring>
using namespace std;
class phiclass
{
    
    
	private:
		int *phiarray,*primearray,philen,primelen;
		bool *lightarray;
	public:
		void init(int *phitmp,int *primetmp,bool *lighttmp,int x);
		int phi(int x);
		int prime(int x);
		bool isprime(int x);
};
void phiclass::init(int *phitmp,int *primetmp,bool *lighttmp,int x)
{
    
    
	int i,j;
	phiarray=phitmp,primearray=primetmp,lightarray=lighttmp,philen=x,primelen=0;
	memset(phiarray,0,sizeof(phiarray));
	memset(primearray,0,sizeof(primearray));
	memset(lightarray,0,sizeof(lightarray));
	phiarray[1]=1,lightarray[0]=lightarray[1]=1;
	for(i=2;i<=philen;i++)
	{
    
    
		if(!lightarray[i])primearray[++primelen]=i,phiarray[i]=i-1;
		for(j=1;j<=primelen&&primearray[j]*i<=philen;j++)
		{
    
    
			lightarray[primearray[j]*i]=1;
			if(i%primearray[j])phiarray[primearray[j]*i]=phiarray[i]*(primearray[j]-1);
			else{
    
    phiarray[primearray[j]*i]=phiarray[i]*primearray[j];break;}
		}
	}
	return;
}
int phiclass::phi(int x)
{
    
    
	return phiarray[x];
}
int phiclass::prime(int x)
{
    
    
	return primearray[x];
}
bool phiclass::isprime(int x)
{
    
    
	return !lightarray[x];
}
int a[500010],b[500010];
bool c[500010];
int main()
{
    
    
	long long n,m,mn,i,ans=0;
	phiclass p;
	for(scanf("%lld%lld",&n,&m),mn=min(n,m),p.init(a,b,c,mn),i=1;i<=mn;i++)ans+=p.phi(i)*(n/i)*(m/i);
	printf("%lld",(ans<<1)-n*m);
	return 0;
}