相关链接
题目传送门:http://www.lydsy.com/JudgeOnline/problem.php?id=4635
数据生成器:http://paste.ubuntu.com/24360378/
神犇题解:http://www.cnblogs.com/clrs97/p/5625508.html
解题报告
对于第一问,显然可以用莫比乌斯反演做到$O(Tm)$
对于第二问,考虑枚举$k$,然后$10^3$以内的数最多有$4$种不同的质因数
于是搞一个状压$DP$,用矩阵快速幂优化
单词询问时间复杂度:$O(32m^2 + 32^3 log (n))$
看起来蛮有希望过的,但卡了一下午常也没有卡过 QwQ
正解的话,我们可以学习Claris用容斥
具体来讲枚举$k$,然后枚举$k$的每一个质因数是否满足条件
然后配合一点预处理之类的,就可以做到$O(m^2 + m \log m)$了
Code
这份代码在BZOJ被卡常了 QwQ
#include<bits/stdc++.h> #define LL long long using namespace std; const int MOD = 1000000007; const int N = 10000009; const int M = 32; int n,m,SZ,s1[N],hc[N],to[1001]; int tot,pri[N>>3],mu[N]; bool vis[N]; inline int read() { char c=getchar(); int f=1,ret=0; while (c<'0'||c>'9') {if(c=='-')f=-1;c=getchar();} while (c<='9'&&c>='0') {ret=ret*10+c-'0';c=getchar();} return ret * f; } inline int Pow(int w, int t) { int ret = 1; for (;t;t>>=1,w=(LL)w*w%MOD) if (t&1) ret=(LL)ret*w%MOD; return ret; } inline void GetMu() { mu[1] = 1; for (int i=2;i<N;i++) { if (!vis[i]) mu[i] = -1, pri[++tot] = i; for (int j=1;j<=tot&&pri[j]*i<N;j++) { vis[i*pri[j]] = 1; if (i%pri[j]) mu[i*pri[j]] = -mu[i]; else {mu[i*pri[j]] = 0; break;} } mu[i] = mu[i-1] + mu[i]; } } inline int cal(int mx) { int ret = 0; for (int l=1,r,tmp;l<=mx;l=r+1) { r = mx / (mx / l); tmp = (LL)(mu[r] - mu[l-1]) * hc[mx/l] % MOD; ret = (ret + tmp) % MOD; } return (ret + MOD) % MOD; } struct Matrix{ int a[M][M]; inline Matrix() {memset(a,0,sizeof(a));} inline Matrix(const Matrix *A) { for (int i=0;i<M;i++) for (int j=0;j<M;j++) a[i][j] = A->a[i][j]; } inline Matrix(int x) { memset(a,0,sizeof(a)); for (int i=0;i<SZ;i++) a[i][i] = x; } inline void operator *= (const Matrix &A) { Matrix ret; for (int i=0,*t1,*t3;i<SZ;i++) { t1 = ret.a[i]; const int *t2 = A.a[i]; for (int j=0;j<SZ;j++) { t3 = a[j]; for (int k=0;k<SZ;k+=4) { t1[k] = (t1[k] + (LL)t3[k] * t2[j]) % MOD; t1[k+1] = (t1[k+1] + (LL)t3[k+1] * t2[j]) % MOD; t1[k+2] = (t1[k+2] + (LL)t3[k+2] * t2[j]) % MOD; t1[k+3] = (t1[k+3] + (LL)t3[k+3] * t2[j]) % MOD; } } } for (int i=0;i<SZ;i++) for (int j=0;j<SZ;j++) a[i][j] = ret.a[i][j]; } inline Matrix operator ^ (int t) { Matrix ret(1),w(this); for (;t;t>>=1,w*=w) if (t&1) ret*=w; return ret; } }ans,trans; inline int solve(int w) { tot = 0; for (int i=2,tmp=w;i<=tmp;i++) { if (tmp % i == 0) { pri[++tot] = i; tmp /= i; while (tmp % i == 0) pri[tot] *= i, tmp /= i; } } ans = Matrix(); trans = Matrix(); ans.a[0][0] = 1; SZ = 1 << tot; memset(to,0,sizeof(to)); for (int i=1,t=1,ww;ww=pri[i],i<=tot;i++,t<<=1) for (int j=ww;j<=m;j+=ww) to[j] |= t; for (int i=1,tt;tt=to[i],i<=m;i++) { for (int p=0,ww;ww=p,p<SZ;p++) ++trans.a[p|tt][p]; } trans = trans ^ n; ans *= trans; return ans.a[SZ-1][0]; } int main() { int T = read(), type = read(); if (type == 1) { GetMu(); for (int t=1,vout;vout=0,t<=T;t++) { n = read(); m = read(); int L = read(), R = read(); for (int l=1,r;l<=m;l=r+1) { r = m / (m / l); hc[m / l] = Pow(m / l, n); } for (int l=L,r,tmp;l<=R;l=r+1) { r = m / (m / l); tmp = cal(m / l); vout = (vout + (min(r,R)-max(L,l)+1ll) * tmp) % MOD; } printf("%d\n",vout); } } else if (type == 2) { for (int t=1,vout,l,r;vout=0,t<=T;t++) { n = read(); m = read(); l = read(), r = read(); for (int w=l;w<=r;w++) vout = (vout + solve(w)) % MOD; printf("%d\n",vout); } } return 0; }
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