CF (439E

Insert n Sweets into an F-1 baffle and divide them into f points (a1, a2, a3... af ). Ask how many methods can make gcd (a1, a2, a3... af) = 1;

Solution; Mobius, first by 1 as the unit, this time there are C (n-1, F-1), and then remove gcd is not 1. At this time, it is required that if the number of quality factors is an odd number, then subtract (mou value is-1), and even numbers are added (mou value is + 1 ), then the mou value of the number of arguments is 0. In this way, we have been able to reprimand. Note that only the number of n-grams is used for calculation. If n-grams are not used, this factor will not appear in gcd.

Code:

/******************************************************* author:xiefubao*******************************************************/#pragma comment(linker, "/STACK:102400000,102400000")#include <iostream>#include <cstring>#include <cstdlib>#include <cstdio>#include <queue>#include <vector>#include <algorithm>#include <cmath>#include <map>#include <set>#include <stack>#include <string.h>//freopen ("in.txt" , "r" , stdin);using namespace std;#define eps 1e-8#define zero(_) (abs(_)<=eps)const double pi=acos(-1.0);typedef long long LL;const int Max=100010;const int INF=1000000007;int mou[Max];LL fac[Max];map<pair<int,int>,LL> maps;int n,f;LL pow(LL a,int b){    LL ans=1;    while(b)    {        if(b&1)            ans=(ans*a)%INF;        a=(a*a)%INF;        b>>=1;    }    return ans;}LL getreverse(LL lo){    return pow(lo,INF-2);}void init(){    for(LL i=2; i<Max; i++)        if(!mou[i])        {            mou[i]=i;            for(LL j=i*i; j<Max; j+=i)                mou[j]=i;               // cout<<i<<" ";        }    mou[1]=1;    for(int i=2; i<Max; i++)    {        if((i/mou[i])%mou[i]==0) mou[i]=0;        else mou[i]=-mou[i/mou[i]];    }    fac[0]=1;    for(int i=1; i<Max; i++)        fac[i]=(fac[i-1]*i)%INF;}LL C(int a,int b){    LL ans=fac[a];    ans=(ans*getreverse(fac[a-b])%INF*getreverse(fac[b]))%INF;    return ans;}int main(){    int t;    cin>>t;    init();    while(t--)    {        scanf("%d%d",&n,&f);        if(maps.find(pair<int,int>(n,f))!=maps.end())        {            printf("%I64d\n",maps[pair<int,int>(n,f)]);            continue;        }        LL ans=0;        for(int i=1; i<=n/f; i++)        {            if((n%i)!=0)continue;            ans+=C(n/i-1,f-1)*mou[i];            if(ans>=INF)                ans-=INF;            if(ans<0)                ans+=INF;        }        printf("%I64d\n",ans);        maps[pair<int,int>(n,f)]=ans;    }    return 0;}