HDU 5976 Detachment 打表找规律

题目链接：

http://acm.hdu.edu.cn/showproblem.php?pid=5976

Detachment

Time Limit: 4000/2000 MS (Java/Others)
Memory Limit: 65536/65536 K (Java/Others)
#### 问题描述
> In a highly developed alien society, the habitats are almost infinite dimensional space.
> In the history of this planet,there is an old puzzle.
> You have a line segment with x units’ length representing one dimension.The line segment can be split into a number of small line segments: a1,a2, … (x= a1+a2+…) assigned to different dimensions. And then, the multidimensional space has been established. Now there are two requirements for this space:
> 1．Two different small line segments cannot be equal ( ai≠aj when i≠j).
> 2．Make this multidimensional space size s as large as possible (s= a1∗a2*...).Note that it allows to keep one dimension.That's to say, the number of ai can be only one.
> Now can you solve this question and find the maximum size of the space?(For the final number is too large,your answer will be modulo 10^9+7)
#### 输入
> The first line is an integer T,meaning the number of test cases.
> Then T lines follow. Each line contains one integer x.
> 1≤T≤10^6, 1≤x≤10^9
#### 输出
> Maximum s you can get modulo 10^9+7. Note that we wants to be greatest product before modulo 10^9+7.
####样例输入
> 1
> 4

样例输出

4

题意

给你一个数n，求把它拆分成若干个数的和，使得这些数的乘积最大。

题解

打表找了下规律：

2

阶段1(3~4)

2 3

阶段2(6~8)

2 3 4

阶段3(10~13)

2 3 4 5

...

处理下阶乘和阶乘的逆，二分找下n属于哪个阶段，然后把区间里面的阶乘拆两部分求就可以了（求x*(x+1)...y利用前缀和思想）。

代码

#include<map>
#include<set>
#include<cmath>
#include<queue>
#include<stack>
#include<ctime>
#include<vector>
#include<cstdio>
#include<string>
#include<bitset>
#include<cstdlib>
#include<cstring>
#include<iostream>
#include<algorithm>
#include<functional>
using namespace std;
#define X first
#define Y second
#define mkp make_pair
#define lson (o<<1)
#define rson ((o<<1)|1)
//#define mid (l+(r-l)/2)
#define sz() size()
#define pb(v) push_back(v)
#define all(o) (o).begin(),(o).end()
#define clr(a,v) memset(a,v,sizeof(a))
#define bug(a) cout<<#a<<" = "<<a<<endl
#define rep(i,a,b) for(int i=a;i<(b);i++)
#define scf scanf
#define prf printf

typedef long long  LL;
typedef vector<int> VI;
typedef pair<int,int> PII;
typedef vector<pair<int,int> > VPII;

const int INF=0x3f3f3f3f;
const LL INFL=0x3f3f3f3f3f3f3f3fLL;
const double eps=1e-6;
const double PI = acos(-1.0);

//start----------------------------------------------------------------------

const int maxn=1e5+10;
const int mod=1e9+7;

LL fac[maxn],invfac[maxn],inv[maxn];
void pre(){
	fac[0]=fac[1]=1;
	invfac[0]=invfac[1]=1;
	inv[1]=1;
	for(int i=2;i<maxn;i++){
		fac[i]=fac[i-1]*i%mod;
		inv[i]=(mod-mod/i)*inv[mod%i]%mod;
		invfac[i]=invfac[i-1]*inv[i]%mod;
	}
}

int n;

inline LL f(LL x){
	return (x+1)*x/2-1;
}

int main() {
	pre();
	int tc;
	scf("%d",&tc);
	while(tc--){
		scf("%d",&n);
		if(n<=4){
			prf("%d\n",n);
			continue;
		}
		int l=2,r=100000;
		while(l+1<r){
			int mid=l+(r-l)/2;
			if(f(mid)<=n) l=mid;
			else r=mid;
		}

		int len=l-1;

		LL ans=1;
		if(f(l+1)-n==1){
			ans=fac[3+len-2]*invfac[2]%mod;
			ans=ans*fac[l+2]%mod*invfac[l+1]%mod;
		}else{
			int d=n-f(l);
			ans=fac[l+1]*invfac[l+1-d]%mod;
			ans=ans*fac[l-d]%mod;
		}

		prf("%lld\n",ans);

	}
    return 0;
}

//end-----------------------------------------------------------------------