[BZOJ3560]DZY Loves Math V(欧拉函数)

https://www.cnblogs.com/zwfymqz/p/9332753.html

由于欧拉函数是积性函数，可以用乘法分配律变成对每个质因子分开算最后乘起来。再由欧拉函数公式和分配律发现就是等比数列求和问题，特判下1的问题就好了。

#include<cstdio>
#include<algorithm>
#define rep(i,l,r) for (int i=(l); i<=(r); i++)
typedef long long ll;
using namespace std;

const int mod=1e9+7;
int n,x,cnt,tot,ans=1,sm[110],p[3510],b[3510];
struct P{ int x,y; }a[800010];
bool cmp(const P &a,const P &b){ return a.x<b.x || (a.x==b.x && a.y<b.y); }

int ksm(int a,int b){
    int res=1;
    for (; b; a=1ll*a*a%mod,b>>=1)
        if (b & 1) res=1ll*res*a%mod;
    return res;
}

void Fac(int x){
    for (int i=1; p[i]*p[i]<=x; i++)
        if (x%p[i]==0){
            int s=0;
            while (x%p[i]==0) x/=p[i],s++;
            a[++cnt]=(P){p[i],s};
        }
    if (x>1) a[++cnt]=(P){x,1};
}

void init(int n){
    rep(i,2,n){
        if (!b[i]) p[++tot]=i;
        for (int j=1; j<=tot && i*p[j]<=n; j++){
            b[i*p[j]]=1;
            if (i%p[j]==0) break;
        }
    }
}

int main(){
    freopen("bzoj3560.in","r",stdin);
    freopen("bzoj3560.out","w",stdout);
    scanf("%d",&n); init(3500);
    rep(i,1,n) scanf("%d",&x),Fac(x);
    sort(a+1,a+cnt+1,cmp);
    for (int i=1,j; i<=cnt; i=j+1){
        for (j=i; j<cnt && a[j+1].x==a[j].x; j++);
        sm[0]=1; int tmp=1;
        rep(k,1,a[j].y) sm[k]=1ll*sm[k-1]*a[i].x%mod;
        rep(k,1,a[j].y) sm[k]=(sm[k-1]+sm[k])%mod;
        rep(k,i,j) tmp=1ll*tmp*sm[a[k].y]%mod;
        tmp=1ll*(tmp-1)*(a[i].x-1)%mod*ksm(a[i].x,mod-2)%mod+1;
        ans=1ll*ans*tmp%mod;
    }
    printf("%d\n",ans);
    return 0;
}