P3768 简单的数学题

题意

首先一波莫比乌斯反演可得（懒得写latex了）：
设\(sum(x)=\sum\limits_{i=1}^xi\)
\(ans=\sum\limits_{T=1}^{n}sum(\frac{n}{T})^2T^2\sum\limits_{d|n}d\mu(\frac{T}{d})\)
有\(id*\mu=\varphi\)
证明：
由\(id=\varphi*1,1*\mu=\epsilon\)可得：
\(id*\mu=\varphi*1*\mu->id*\mu=\varphi\)
知道这个后：
\(\sum\limits_{T=1}^{n}sum(\frac{n}{T})^2T^2\varphi(T)\)
显然前面可以除法分块，考虑怎么求\(T^2\varphi(T)\)

设\(f(i)=i^2*\varphi(i)\)，发现这是个积性函数，考虑杜教筛：
设\(S(n)=\sum\limits_{i=1}^{n}f(i)\)
先上杜教筛套路式子：
\(g(1)*S(n)=\sum\limits_{i=1}^n(g*f)(i)-\sum\limits_{i=2}^ng(i)*f(\frac{n}{i})\)
考虑找合适的\(g\)：
\((f*g)(n)=\sum\limits_{d|n}g(d)*f(\frac{n}{d})\)
\(=\sum\limits_{d|N}g(d)*\frac{n^2}{d^2}\varphi(\frac{n}{d})\)
由于\(\sum\limits_{d|n}\varphi(d)=n\)，所以令\(g(x)=x^2\)
\(=\sum\limits_{d|n}d^2*\frac{n^2}{d^2}\varphi(\frac{n}{d})\)
\(=\sum\limits_{d|n}n^2\varphi(\frac{n}{d})\)
\(=n^2\sum\limits_{d|n}\varphi(\frac{n}{d})\)
\(=n^3\)
因此当\(g(x)=x^2\)时，\((f*g)(x)=x^3\)。

于是先杜教筛再除法分块即可。

code:

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int maxn=8*1e6+10;
ll n,mod,inv2,inv6,ans;
ll phi[maxn],sum[maxn];
bool vis[maxn];
vector<int>prime;
unordered_map<ll,ll>mp;
inline ll sqr(ll x){return x*x%mod;}
inline ll calc1(ll x){x%=mod;return x*(x+1)%mod*(2*x+1)%mod*inv6%mod;}
inline ll calc2(ll x){x%=mod;return x*(x+1)%mod*inv2%mod;}
inline ll power(ll x,ll k,ll mod)
{
    ll res=1;
    while(k)
    {
        if(k&1)res=res*x%mod;
        x=x*x%mod;k>>=1;
    }
    return res;
}
inline void pre_work(int n)
{
    vis[1]=1;phi[1]=1;
    for(int i=2;i<=n;i++)
    {
        if(!vis[i])prime.push_back(i),phi[i]=(i-1)%mod;
        for(unsigned int j=0;j<prime.size()&&i*prime[j]<=n;j++)
        {
            vis[i*prime[j]]=1;
            if(i%prime[j]==0)
            {
                phi[i*prime[j]]=1ll*phi[i]*prime[j]%mod;
                break;
            }
            phi[i*prime[j]]=1ll*phi[i]*phi[prime[j]]%mod;           
        }
    }
    for(int i=1;i<=n;i++)sum[i]=(sum[i-1]+1ll*phi[i]*i%mod*i%mod)%mod;
}
inline ll getsum(ll x)
{
    if(x<=8000000)return sum[x];
    if(mp.count(x))return mp[x];
    ll res=sqr(calc2(x));
    for(ll l=2,r;l<=x;l=r+1)
    {
        r=x/(x/l);
        res-=(calc1(r)-calc1(l-1))%mod*getsum(x/l)%mod;
        res%=mod;
    }
    return mp[x]=(res+mod)%mod;
}
int main()
{
    scanf("%lld%lld",&mod,&n);
    inv2=power(2,mod-2,mod),inv6=power(6,mod-2,mod);
    pre_work(8000000);
    for(ll l=1,r;l<=n;l=r+1)
    {
        r=n/(n/l);
        ans=(ans+sqr(calc2(n/l))*(getsum(r)-getsum(l-1))%mod+mod)%mod;
    }
    printf("%lld",(ans+mod)%mod);
    return 0;
}