Easy Math 問答問題反饋 只看題面

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Given a positive integers?nn?, Mobius function?\mu(n)μ(n)?is defined as follows:

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\displaystyle \mu(n) = \begin{cases} 1 &n = 1 \\ (-1)^k & n = p_1p_2\cdots p_k \\ 0 &other \end{cases}μ(n)=??????1(?1)k0?n=1n=p1?p2??pk?other?

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p_i (i = 1, 2, \cdots, k)pi?(i=1,2,?,k)?are different prime numbers.

Given two integers?mm,?nn, please calculate?\sum_{i = 1}^{m} \mu(in)∑i=1m?μ(in).

Input

One line includes two integers?m (1 \le m \le 2e9)m(1≤m≤2e9),?n (1 \le n \le 1e12)n(1≤n≤1e12).

Output

One line includes the answer .

樣例輸入復(fù)制

2 2

樣例輸出復(fù)制

-1

題目來源

ACM-ICPC 2018 徐州賽區(qū)網(wǎng)絡(luò)預(yù)賽

min_25篩的話，先得到下面這個公式

首先答案可以化簡為這個

根據(jù)積性函數(shù)前綴和可以得到這個

(S_f(m,1)+1)*mu(n)

#include<bits/stdc++.h> using namespace std; #define ll long long const int N=1e6+5; ll p[N],w[N],pri[N],g[N],M,m,n; int block,id[N],sz,tot; bool is_p[N],v[N]; void init(int n) {sz=0;for(int i=2; i<=n; i++){if(!is_p[i])p[++sz]=i;for(int j=1; j<=sz&&p[j]*i<n; j++){is_p[i*p[j]]=1;if(i%p[j]==0) break;}} } void sieve_g(ll n) {M=0;for(ll i=1,lst; i<=n; i=lst+1){ll len=n/i;lst=n/len;w[++M]=len;g[M]=1-w[M];if(len<=block) id[len]=M;}for(int i=1; i<=sz; i++)for(int j=1; j<=M&&p[i]*p[i]<=w[j]; j++){int op=w[j]/p[i]<=block?id[w[j]/p[i]]:n/(w[j]/p[i]);g[j]-=g[op]+i-1;} } ll S(ll x,ll y) {ll k,ans=0;if(x<=1||p[y]>x) return 0;if(x>block) k=m/x;else k=id[x];ans=g[k]+y-1;for(int i=1; i<=tot&&pri[i]<=w[k]; i++)if(pri[i]>p[y-1]) ans++;for(int i=y; i<=sz&&p[i]*p[i]<=x; i++)if(v[i]==0) ans-=S(x/p[i],i+1);return ans; } int main() {tot=0;cin>>m>>n;block=sqrt(m+0.5);init(m<=1e6?1e4:1e6);sieve_g(m);for(int i=1; p[i]*p[i]<=n&&i<=sz;i++){if(n%p[i])continue;pri[++tot]=p[i];n/=p[i];v[i]=1;if(n%p[i]==0){puts("0");return 0;}}if(n>1){pri[++tot]=n;for(int i=sz; i>0; i--)if(p[i]==n){v[i]=1;break;}}int t=tot&1?-1:1;cout<<(S(m,1)+1)*t;return 0; }

莫比烏斯+杜教篩

#include<bits/stdc++.h> using namespace std; #define ll long long const int N=1e6+5; int pri[N],num; int vst[N],mu[N],mu_n; inline void Pre() {mu[1]=1;for(int i=2; i<N; i++){if(!vst[i]) pri[++num]=i,mu[i]=-1;for(int j=1; j<=num && i*1LL*pri[j]<N; j++){vst[i*pri[j]]=1;if(i%pri[j]==0){mu[i*pri[j]]=0;break;}mu[i*pri[j]]=mu[i]*mu[pri[j]];}}for(int i=1; i<N; i++) mu[i]+=mu[i-1]; } unordered_map<ll,int> S; inline int Sum(ll n) {if(n<N) return mu[n];if(S.count(n)) return S[n];int tem=1;ll l,r;for(l=2; l*l<=n; l++) tem-=Sum(n/l);for(ll t=n/l; l<=n; l=r+1,t--){r=n/t;tem-=(r-l+1)*Sum(t);}return S[n]=tem; } ll f(ll m,ll n) {if(m==0) return 0;if(n==1) return Sum(m);int flag=1;for(int i=1; pri[i]*1LL*pri[i]<=n; i++){if(n%pri[i]==0){flag=0;return -f(m,n/pri[i])+f(m/pri[i],n);}}if(flag)return -f(m,n/n)+f(m/n,n); } int judge(ll n) {for(int i=1; pri[i]*1LL*pri[i]<=n; i++){if(n%pri[i]==0){int cnt=0;while(n%pri[i]==0)n/=pri[i],cnt++;if(cnt>=2)return 0;}}return 1; } int main() {Pre();ll m,n;cin>>m>>n;if(!judge(n)){printf("0\n");return 0;}cout<<f(m,n);;return 0; }

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轉(zhuǎn)載于:https://www.cnblogs.com/BobHuang/p/9702972.html

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