P3455 [POI2007]ZAP-Queries

题目描述

Byteasar the Cryptographer works on breaking the code of BSA (Byteotian Security Agency). He has alreadyfound out that whilst deciphering a message he will have to answer multiple queries of the form"for givenintegers aaa, bbb and ddd, find the number of integer pairs (x,y)(x,y)(x,y) satisfying the following conditions:

1≤x≤a1\le x\le a1≤x≤a,1≤y≤b1\le y\le b1≤y≤b,gcd(x,y)=dgcd(x,y)=dgcd(x,y)=d, where gcd(x,y)gcd(x,y)gcd(x,y) is the greatest common divisor of xxx and yyy".

Byteasar would like to automate his work, so he has asked for your help.

TaskWrite a programme which:

reads from the standard input a list of queries, which the Byteasar has to give answer to, calculates answers to the queries, writes the outcome to the standard output.

FGD正在破解一段密码，他需要回答很多类似的问题：对于给定的整数a,b和d，有多少正整数对x,y，满足x<=a，y<=b，并且gcd(x,y)=d。作为FGD的同学，FGD希望得到你的帮助。

输入输出格式

输入格式：

The first line of the standard input contains one integer nnn (1≤n≤50 0001\le n\le 50\ 0001≤n≤50 000),denoting the number of queries.

The following nnn lines contain three integers each: aaa, bbb and ddd(1≤d≤a,b≤50 0001\le d\le a,b\le 50\ 0001≤d≤a,b≤50 000), separated by single spaces.

Each triplet denotes a single query.

输出格式：

Your programme should write nnn lines to the standard output. The iii'th line should contain a single integer: theanswer to the iii'th query from the standard input.

输入输出样例

输入样例#1：

输出样例#1：

3

2

Solution：

　　本题莫比乌斯反演板子题。

　　题意就是求$\sum_\limits{i=1}^{i\leq n}\sum_\limits{j=1}^{j\leq m} gcd(i,j)==d$。

　　令$f(n)$表示满足约束条件的且$gcd(i,j)=n$的个数，令$F(n)$表示满足约束条件的且$n|gcd(i,j)$的个数。

　　于是有$F(n)=\sum_\limits{n|d} {f(d)}$，这个式子显然可以反演，得到$f(n)=\sum_\limits{n|d} {\mu(\frac{d}{n})*F(d)}$。

　　又因为对于数$x$，显然$n$中含有$\frac{n}{x}$个$x$的倍数，同理$m$中有$\frac{m}{x}$个，所以$F(x)=\frac{n}{x}*\frac{m}{x}$。

　　则原式变为$f(d)=\sum_\limits{d|k}^{k\leq min(n,m)}{\mu(\frac{k}{d})\frac{n}{k}\frac{m}{k}}$。

　　令$t=\frac{k}{d}$，则原式变为$f(d)=\sum_\limits{t=1}^{\frac{min(n,m)}{d}}{\frac{n}{td}\frac{m}{td}}$，对于$\lfloor \frac{n}{td}\rfloor \lfloor \frac{m}{td} \rfloor$直接数论分块求就好了，所以还得预处理下$\mu$的前缀和。

　　时间复杂度$O(q\sqrt n)$。

代码：

/*Code by 520 -- 9.10*/

#include<bits/stdc++.h>

#define il inline

#define ll long long

#define RE register

#define For(i,a,b) for(RE int (i)=(a);(i)<=(b);(i)++)

#define Bor(i,a,b) for(RE int (i)=(b);(i)>=(a);(i)--)

using namespace std;

const int N=;

int n,a,b,d,mu[N],prime[N],cnt;

bool isprime[N];

int gi(){

    int a=;char x=getchar();

    while(x<''||x>'')x=getchar();

    while(x>=''&&x<='')a=(a<<)+(a<<)+(x^),x=getchar();

    return a;

}

il void pre(){

    mu[]=;

    For(i,,){

        if(!isprime[i]) mu[i]=-,prime[++cnt]=i;

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

            isprime[i*prime[j]]=;

            if(i%prime[j]==) break;

            mu[i*prime[j]]=-mu[i];

        }

    }

    For(i,,) mu[i]+=mu[i-];

}

int solve(){

    if(a>b) swap(a,b);

    a/=d,b/=d;

    int pos=,ans=;

    for(int i=;i<=a;i=pos+){

        pos=min(a/(a/i),b/(b/i));

        ans+=(mu[pos]-mu[i-])*(a/i)*(b/i);

    }

    return ans;

}

int main(){

    pre();

    n=gi();

    while(n--){

        a=gi(),b=gi(),d=gi();

        printf("%d\n",solve());

    }

    return ;

}