hdu 5382

\(F(n)=\sum_{i=1}^n\sum_{j=1}^n[lcm(i,j)+gcd(i,j)\geq n]\)

\(S(n)=\sum_{i=1}^nF(i)\)

\(F(n)=n^2-\sum_{i=1}^n\sum_{j=1}^n[lcm(i,j)+gcd(i,j)<n]\)

\(F(n)-F(n-1)=n^2-\sum_{i=1}^n\sum_{j=1}^n[lcm(i,j)+gcd(i,j)<n]-(n-1)^2+\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}[lcm(i,j)+gcd(i,j)<n-1]\)

\(F(n)-F(n-1)=2*n-1-\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}[lcm(i,j)+gcd(i,j)=n-1]\)

\(g(n)=\sum_{i=1}^n\sum_{j=1}^n[lcm(i,j)+gcd(i,j)=n]\)

\(g(n)=\sum_{d|n}\sum_{i=1}^{\frac{n}{d}}\sum_{j=1}^{\frac{n}{d}}[gcd(i,j)=1][i*j=\frac{n}{d}-1]\)

\(G(n)=\sum_{i=1}^n\sum_{j=1}^n[gcd(i,j)=1][i*j=n-1]\)

\(G(n)=\sum_{i=1}[gcd(i,\frac{n}{i})=1]\)

\(g(n)=\sum_{d|n}G(d)\)

\(F(n)=F(n-1)+2*n-1-g(n-1)\)

//#pragma GCC optimize(2)

//#pragma GCC optimize(3)

//#pragma GCC optimize(4)

//#pragma GCC optimize("unroll-loops")

//#pragma comment(linker, "/stack:200000000")

//#pragma GCC optimize("Ofast,no-stack-protector")

//#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")

#include<bits/stdc++.h>

//#include <bits/extc++.h>

#define fi first

#define se second

#define db double

#define mp make_pair

#define pb push_back

#define mt make_tuple

//#define pi acos(-1.0)

#define ll long long

#define vi vector<int>

#define mod 258280327

#define ld long double

//#define C 0.5772156649

#define ls l,m,rt<<1

#define rs m+1,r,rt<<1|1

#define sqr(x) ((x)*(x))

#define pll pair<ll,ll>

#define pil pair<int,ll>

#define pli pair<ll,int>

#define pii pair<int,int>

#define ull unsigned long long

#define bpc __builtin_popcount

#define base 1000000000000000000ll

#define fin freopen("a.txt","r",stdin)

#define fout freopen("a.txt","w",stdout)

#define fio ios::sync_with_stdio(false);cin.tie(0)

#define mr mt19937 rng(chrono::steady_clock::now().time_since_epoch().count())

inline ll gcd(ll a,ll b){return b?gcd(b,a%b):a;}

inline void sub(ll &a,ll b){a-=b;if(a<0)a+=mod;}

inline void add(ll &a,ll b){a+=b;if(a>=mod)a-=mod;}

template<typename T>inline T const& MAX(T const &a,T const &b){return a>b?a:b;}

template<typename T>inline T const& MIN(T const &a,T const &b){return a<b?a:b;}

inline ll mul(ll a,ll b,ll c){return (a*b-(ll)((ld)a*b/c)*c+c)%c;}

inline ll qp(ll a,ll b){ll ans=1;while(b){if(b&1)ans=ans*a%mod;a=a*a%mod,b>>=1;}return ans;}

inline ll qp(ll a,ll b,ll c){ll ans=1;while(b){if(b&1)ans=mul(ans,a,c);a=mul(a,a,c),b>>=1;}return ans;}

using namespace std;

//using namespace __gnu_pbds;

const ld pi=acos(-1);

const ull ba=233;

const db eps=1e-5;

const ll INF=0x3f3f3f3f3f3f3f3f;

const int N=1000000+10,maxn=2000000+10,inf=0x3f3f3f3f;

ll f[N],g[N],F[N];

int prime[N],cnt;

bool mark[N];

void init()

{

    for(int i=2;i<N;i++)

    {

        if(!mark[i])prime[++cnt]=i,f[i]=1;

        for(int j=1;j<=cnt&&i*prime[j]<N;j++)

        {

            mark[i*prime[j]]=1;

            if(i%prime[j]==0)

            {

                f[i*prime[j]]=f[i];

                break;

            }

            f[i*prime[j]]=f[i]+1;

        }

    }

    for(int i=1;i<N;i++)f[i]=qp(2,f[i]);

    for(int i=1;i<N;i++)

        for(int j=i;j<N;j+=i)

            add(g[j],f[j/i-1]);

    for(int i=1;i<N;i++)F[i]=((F[i-1]+2*i-1-g[i-1])%mod+mod)%mod;

    for(int i=1;i<N;i++)add(F[i],F[i-1]);

}

int main()

{

    init();

    int t;scanf("%d",&t);

    while(t--)

    {

        int x;scanf("%d",&x);

        printf("%lld\n",F[x]);

    }

    return 0;

}

/********************

********************/

巴特西

hdu 5382

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