1853: [Scoi2010]幸运数字[容斥原理]

1853: [Scoi2010]幸运数字

Time Limit: 2 Sec  Memory Limit: 64 MB
Submit: 2405  Solved: 887
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Description

在中国，很多人都把6和8视为是幸运数字！lxhgww也这样认为，于是他定义自己的“幸运号码”是十进制表示中只包含数字6和8的那些号码，比如68，666，888都是“幸运号码”！但是这种“幸运号码”总是太少了，比如在[1,100]的区间内就只有6个（6，8，66，68，86，88），于是他又定义了一种“近似幸运号码”。lxhgww规定，凡是“幸运号码”的倍数都是“近似幸运号码”，当然，任何的“幸运号码”也都是“近似幸运号码”，比如12，16，666都是“近似幸运号码”。 现在lxhgww想知道在一段闭区间[a, b]内，“近似幸运号码”的个数。

Input

输入数据是一行，包括2个数字a和b

Output

输出数据是一行，包括1个数字，表示在闭区间[a, b]内“近似幸运号码”的个数

Sample Input

【样例输入1】
1 10
【样例输入2】
1234 4321

Sample Output

【样例输出1】
2
【样例输出2】
809

HINT

【数据范围】
对于30%的数据，保证1 < =a < =b < =1000000
对于100%的数据，保证1 < =a < =b < =10000000000

Source

Day1

/*
    一个非常显然的事实就是幸运号码不会太多。把表打出来，就会发现在10^10以内的幸运数只有2404个
　　这个时候一个非常显然的想法就是对这些数进行容斥，即加上每个数的倍数个数，减去两个数的lcm的倍数个数，加上三个的lcm的倍数个数，……以此类推。
　　这样的复杂度显然是不对的，理论上可达O(2^x)，其中xx为幸运数个数。但是由于多个数的倍数不能超过右边界，就可以剪枝，实际复杂度低了不知道多少。
　　但是这样时间不够。我们还可以对幸运数进行处理，将其中是另外的幸运数的倍数的数给去掉。这样可以将需要考虑的数的个数减掉一半左右。
　　最后还有一个小优化，那就是将最后需要处理的幸运数按从大到小排好序。这样可以让乘积尽早变得更大，可以减掉许多不必要的计算。
　　加了上述优化，就差不多可以AC了。
*/
#include<cstdio>
#include<iostream>
#include<algorithm>
#define set(x) freopen(#x".in","r",stdin);freopen(#x".out","w",stdout);
using namespace std;
const int N=3000;
typedef long long ll;
ll l,r,now,ci,ans;
ll num,cnt,a[N],b[N];
void init(ll x){
    if(x>r) return ;
    a[num++]=x;
    init(x*10+6);
    init(x*10+8);
}
void dfs(ll x){
    if(x>cnt){
        if(!ci) return ;
        if(ci&1) ans+=r/now-(l-1)/now;
        else ans-=r/now-(l-1)/now;
        return ;
    }
    dfs(x+1);
    ll tx=now;
    now=now/__gcd(now,b[x]);ci++;
    if((double)now*b[x]<=r){
        now*=b[x];
        dfs(x+1);
    }
    ci--;now=tx;
}
int main(){
    set(luckynumber);
    cin>>l>>r;
    init(0);num--;
    sort(a+1,a+num+1);
    for(int i=1;i<=num;i++){
        b[++cnt]=a[i];
        for(int j=1;j<cnt;j++){
            if(!(a[i]%b[j])){
                cnt--;break;
            }
        }
    }
    reverse(b+1,b+cnt+1);
    now=1;
    dfs(1);
    cout<<ans;
    return 0;
}
/*30分hash 
#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#define set(x) freopen(#x".in","r",stdin);freopen(#x".out","w",stdout);
using namespace std;
typedef long long ll;
ll a,b,n;
ll 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struct Thash{
    static const int N=2e6+5,MOD=1733333;
    ll tot,head[MOD+10],val[N],h[N],next[N];
    void clear(){tot=0;memset(head,0,sizeof head);}
    void insert(ll H,ll VAL){
        for(ll i=head[H%MOD];i;i=next[i]) if(h[i]==H){val[i]=VAL;return ;}
        h[++tot]=H;val[tot]=VAL;next[tot]=head[H%MOD];head[H%MOD]=tot;
    }
    ll get(ll H){
        for(ll i=head[H%MOD];i;i=next[i]) if(h[i]==H) return val[i];
        return 0;
    }
}M;
ll solve(ll lim){
    ll ans=0;
    M.clear();
    for(ll i=0;;i++){
        if(list[i]>lim) break;
        for(ll j=list[i];j<=lim;j+=list[i]){
            if(!M.get(j)){
                ans++;
                M.insert(j,1);
            }
        }
    }
    return ans;
}
int main(){
    set(luckynumber);
    cin>>a>>b;
    cout<<solve(b)-solve(a-1);
    return 0;
}*/