bzoj 2956 模积和

2019-04-13 17:29发布生成海报

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　求∑∑((n mod i)*(m mod j))其中1<=i<=n,1<=j<=m,i≠j。这份代码调试了很久..菜把式子随便推导一下展开即可注意取模 6的逆元直接exgcd算完公式:

\sum_{i = 1}^{n} \sum_{j = 1}^{m} (n \mod i) \times (m \mod j) - \sum_{i = 1}^{m i n (n, m)} (n \mod i) \times (m \mod j)

\sum_{i = 1}^{n} \sum_{j = 1}^{m} (n - i \times ⌊ \frac{n}{i} ⌋) \times (m - j \times ⌊ \frac{m}{j} ⌋) - \sum_{i = 1}^{m i n (n, m)} (n - i \times ⌊ \frac{n}{i} ⌋) \times (m - j \times ⌊ \frac{m}{j} ⌋)

\sum_{i = 1}^{n} (n - i \times ⌊ \frac{n}{i} ⌋) \sum_{j = 1}^{m} (m - j \times ⌊ \frac{m}{j} ⌋) - \sum_{i = 1}^{m i n (n, m)} (n - i \times ⌊ \frac{n}{i} ⌋) \times (m - j \times ⌊ \frac{m}{j} ⌋))

#include
#include
#define ll long long
using namespace std;
const int mod=19940417;
const int inv=3323403;
inline int inc(int x,int v){return x+v>=mod?x+v-mod:x+v;}
inline int dec(int x,int v){return x-v<0?x-v+mod:x-v;}
inline int gao(int n,int m){
    int tmp=0,last;
    for (int i=1;i<=n;i=last+1){
        last=m/(m/i);if (last>n) last=n;
        tmp=inc(tmp,(ll)(m/i)*((ll)(last-i+1)*(last+i)/2%mod)%mod);
    }return tmp;
}
inline int calc(int x){return (ll)x*(x+1)%mod*(2*x+1)%mod*inv%mod;}
int n,m,last; 
int main(){
    freopen("bzoj2956.in","r",stdin);
    scanf("%d%d",&n,&m);int ans=0,ans1=0,ans2=0;
    ans1=inc(ans1,(ll)n*n%mod);ans2=inc(ans2,(ll)m*m%mod);
    ans1=dec(ans1,gao(n,n));ans2=dec(ans2,gao(m,m));
    if (n>m) swap(n,m);ans=inc(ans,(ll)n*m%mod*n%mod);
    for (int i=1;i<=n;i=last+1){
        last=min(n/(n/i),m/(m/i));
        ans=inc(ans,(ll)(m/i)*(n/i)%mod*dec(calc(last),calc(i-1))%mod);
    }
//  printf("%d
",m*gao(n,n));
//  printf("%d
",n*gao(n,m));
    ans=dec(ans,(ll)m*gao(n,n)%mod);ans=dec(ans,(ll)n*gao(n,m)%mod);
    ans1=dec((ll)ans1*ans2%mod,ans);printf("%d
",ans1);
    return 0;
}

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