【题解】Luogu P2257 YY的GCD

た入场券 2021-10-01 06:44 297阅读 0赞

### [原题传送门][Link 1] ###

### 这题需要运用[莫比乌斯反演(懵逼钨丝繁衍)][Link 2] ###

#### 显然题目的答案就是\\\[ Ans=\\sum\_\{i=1\}^N\\sum\_\{j=1\}^M\[gcd(i,j)=prime\]\\\] ####

#### 我们先设设F(n)表示满足\\(gcd(i,j)\\%t=0\\)的数对个数，f(t)表示满足\\(gcd(i,j)=t\\)的数对个数 ####

#### \\\[f(t)=\\sum\_\{i=1\}^N\\sum\_\{j=1\}^M\[gcd(i,j)=t\]\\\] ####

#### \\\[F(n)=\\sum\_\{n|t\}\\lfloor \\frac\{N\}\{n\} \\rfloor \\lfloor \\frac\{M\}\{n\} \\rfloor\\\] ####

#### 那么根据莫比乌斯反演的第二种形式珂以得到 ####

#### \\\[f(n)=\\sum\_\{n|t\}\\mu(\\lfloor \\frac\{t\}\{n\} \\rfloor)F(t)\\\] ####

#### 所以答案珂以变形为： ####

#### \\\[Ans=\\sum\_\{p \\in prime\}\\sum\_\{i=1\}^N\\sum\_\{j-1\}^M\[gcd(i,j)=p)\\\] ####

#### \\\[=\\sum\_\{p \\in prime\}f(p) \\qquad \\qquad \\quad\\\] ####

#### \\\[=\\sum\_\{p \\in prime\}\\sum\_\{p|t\}\\mu(\\lfloor \\frac\{t\}\{p\} \\rfloor)F(t)\\\] ####

#### 我们不枚举p，我们枚举\\(\\lfloor \\frac\{t\}\{p\} \\rfloor\\) ####

#### \\\[Ans=\\sum\_\{p \\in prime\}\\sum\_\{d=1\}^\{Min(\\frac\{N\}\{p\},\\frac\{M\}\{p\})\}\\mu(t)F(tp)\\\] ####

#### \\\[\\qquad \\qquad \\qquad=\\sum\_\{p \\in prime\}\\sum\_\{d=1\}^\{Min(\\frac\{N\}\{p\},\\frac\{M\}\{p\})\}\\mu(t)\\sum\_\{n|t\}\\lfloor \\frac\{N\}\{tp\} \\rfloor \\lfloor \\frac\{M\}\{tp\} \\rfloor\\\] ####

#### 我们把tp换成T继续变形 ####

#### \\\[Ans=\\sum\_\{T=1\}^\{Min(N,M)\}\\lfloor \\frac\{N\}\{T\} \\rfloor \\lfloor \\frac\{M\}\{T\} \\rfloor(\\sum\_\{p|T,p \\in prime\}\\mu(\\frac\{T\}\{p\}))\\\] ####

#### 这样就珂以用整除分块求了qaq ####

#include <bits/stdc++.h> #define N 10000005 #define ll long long #define getchar nc using namespace std; inline char nc(){ static char buf[100000],*p1=buf,*p2=buf; return p1==p2&&(p2=(p1=buf)+fread(buf,1,100000,stdin),p1==p2)?EOF:*p1++; } inline int read() { register int x=0,f=1;register char ch=getchar(); while(ch<'0'||ch>'9'){if(ch=='-')f=-1;ch=getchar();} while(ch>='0'&&ch<='9')x=(x<<3)+(x<<1)+ch-'0',ch=getchar(); return x*f; } inline void write(register ll x) { if(!x)putchar('0');if(x<0)x=-x,putchar('-'); static int sta[30];register int tot=0; while(x)sta[tot++]=x%10,x/=10; while(tot)putchar(sta[--tot]+48); } inline int Min(register int x,register int y) { return x<y?x:y; } int v[N],miu[N],prim[N],cnt=0,g[N]; ll sum[N]; int main() { miu[1]=1; for(register int i=2;i<=N;++i) { if(!v[i]) { miu[i]=-1; prim[++cnt]=i; } for(register int j=1;j<=cnt&&prim[j]*i<=N;++j) { v[i*prim[j]]=1; if(i%prim[j]==0) break; else miu[prim[j]*i]=-miu[i]; } } for(register int i=1;i<=cnt;++i) for(register int j=1;j*prim[i]<=N;++j) g[j*prim[i]]+=miu[j]; for(register int i=1;i<=N;++i) sum[i]=sum[i-1]+(ll)g[i]; int t=read(); while(t--) { int n=read(),m=read(); if(n>m) n^=m^=n^=m; ll ans=0; for(register int l=1,r;l<=n;l=r+1) { r=Min(n/(n/l),m/(m/l)); ans+=(ll)(n/l)*(m/l)*(sum[r]-sum[l-1]); } write(ans),puts(""); } return 0; }

转载于:https://www.cnblogs.com/yzhang-rp-inf/p/10134439.html

[Link 1]: https://www.luogu.org/problemnew/show/P4450
[Link 2]: https://www.cnblogs.com/yzhang-rp-inf/p/10124488.html