(Description)
求$$sum_{i-1}^nsum_{j=1}^nijgcd(i,j)mod p$$
(n<=10^{10})
(Solution)
[sum_{i-1}^nsum_{j=1}^nijgcd(i,j)
]
[=sum_{d=1}^nd^3sum_{i=1}^{n/d}sum_{j=1}^{n/d}ij[gcd(i,j)==1]
]
发现跟这道题只差了一个(d^3)。最后化简得到
[ans=sum_{T=1}^n(frac{frac{n}{T} imes(frac{n}{T}+1)}{2})^2phi(T)T^2
]
令(f(i)=phi(i)i^2)。
已知(phi*1=Id),所以可以令(g(i)=i^2),这样(f*g(x)=sum_{d|x}phi(d)d^2frac{x^2}{d^2}=x^3),(f*g)的前缀和就可以(O(1)计算了)。
#include<complex>
#include<cstdio>
#include<map>
#define LL long long
using namespace std;
const int N=7e6+7;
int tot,ans,mod,nn,div6;
LL n;
int prime[N],phi[N];
bool check[N];
map<LL,int>mp;
void Init()
{
check[1]=phi[1]=1;
nn=min(n,(LL)N-1);
for(int i=2;i<=nn;i++)
{
if(!check[i])prime[++tot]=i,phi[i]=i-1;
for(int j=1;j<=tot && i*prime[j]<=nn;j++)
{
check[i*prime[j]]=1;
if(i%prime[j])phi[i*prime[j]]=phi[i]*phi[prime[j]];
else
{
phi[i*prime[j]]=phi[i]*prime[j];
break;
}
}
}
for(int i=1;i<=nn;i++)
phi[i]=(phi[i-1]+1ll*i*i%mod*phi[i]%mod)%mod;
}
int calc1(LL x)
{
x%=mod;
return x*(x+1)/2%mod;
}
int calc2(LL x)
{
x%=mod;
return x*(x+1)%mod*(x+x+1)%mod*div6%mod;
}
LL Sum(LL x)
{
if(x<=nn)return phi[x];
if(mp[x])return mp[x];
LL res=calc1(x);res=res*res%mod;
for(LL l=2,r;l<=x;l=r+1)
{
r=x/(x/l);
res=(res-(calc2(r)-calc2(l-1))*Sum(x/l)%mod)%mod;
}
return mp[x]=(res+mod)%mod;
}
int Fpow(LL b,int p)
{
LL res=1;
for(;p;p>>=1,b=b*b%mod)
if(p&1)res=res*b%mod;
return res;
}
int main()
{
scanf("%d%lld",&mod,&n);
div6=Fpow(6,mod-2);
Init();
for(LL l=1,r;l<=n;l=r+1)
{
r=n/(n/l);
LL tmp=calc1(n/l);
ans=(ans+tmp*tmp%mod*(Sum(r)-Sum(l-1))%mod)%mod;
}
printf("%d
",(ans+mod)%mod);
return 0;
}