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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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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;}*/

 

1853: [Scoi2010]幸运数字[容斥原理]