SPOJ 20713 DIVCNT2 - Counting Divisors (square)

DIVCNT2 - Counting Divisors (square)

#sub-linear #dirichlet-generating-function

 

Let \sigma_0(n)σ​0​​(n) be the number of positive divisors of nn.

For example, \sigma_0(1) = 1σ​0​​(1)=1, \sigma_0(2) = 2σ​0​​(2)=2 and \sigma_0(6) = 4σ​0​​(6)=4.

LetS_2(n) = \sum _{i=1}^n \sigma_0(i^2).S​2​​(n)=​i=1​∑​n​​σ​0​​(i​2​​).

Given NN, find S_2(N)S​2​​(N).

Input

First line contains TT (1 \le T \le 100001≤T≤10000), the number of test cases.

Each of the next TT lines contains a single integer NN. (1 \le N \le 10^{12}1≤N≤10​12​​)

Output

For each number NN, output a single line containing S_2(N)S​2​​(N).

Example

Input

5
1
2
3
10
100

Output

1
4
7
48
1194

Explanation for Input

- S_2(3) = \sigma_0(1^2) + \sigma_0(2^2) + \sigma_0(3^2) = 1 + 3 + 3 = 7S​2​​(3)=σ​0​​(1​2​​)+σ​0​​(2​2​​)+σ​0​​(3​2​​)=1+3+3=7

Information

There are 6 Input files.

- Input #1: 1 \le N \le 100001≤N≤10000, TL = 1s.

- Input #2: 1 \le T \le 800,\ 1 \le N \le 10^{8}1≤T≤800, 1≤N≤10​8​​, TL = 20s.

- Input #3: 1 \le T \le 200,\ 1 \le N \le 10^{9}1≤T≤200, 1≤N≤10​9​​, TL = 20s.

- Input #4: 1 \le T \le 40,\ 1 \le N \le 10^{10}1≤T≤40, 1≤N≤10​10​​, TL = 20s.

- Input #5: 1 \le T \le 10,\ 1 \le N \le 10^{11}1≤T≤10, 1≤N≤10​11​​, TL = 20s.

- Input #6: T = 1,\ 1 \le N \le 10^{12}T=1, 1≤N≤10​12​​, TL = 20s.

My C++ solution runs in 5.3 sec. (total time)

Source Limit is 6 KB.

 

 

很迷的函数题。

如何求 d(i^2)?

d(i^2)= (2*a1+1)(2*a2+1)(2*a3+1)...(2*ak+1)

我们考虑一下选哪些质因子的集合，上式

=Σ2^|S| *π a[i] ,i属于S

=Σ(p|i)  2^w(p)。

其中w(x)为x的质因子数。

然后发现2^w(x)=Σ(i|x)  μ^2(i)

 

所以ANS= Σμ^2(i) *Σd(j)  ,其中1<=i<=n,1<=j<=(n/i)。

#include<bits/stdc++.h>
#define ll long long
using namespace std;
int zs[10000005],t=0,T,sq[50000005];
int miu[50000005],low[50000005],maxn;
bool v[50000005];
ll d[50000005],n;

inline void init(){
	miu[1]=1,d[1]=1,low[1]=1;
	for(int i=2;i<=maxn;i++){
		if(!v[i]) zs[++t]=i,miu[i]=-1,d[i]=2,low[i]=i;
		for(int j=1,u;j<=t&&(u=zs[j]*i)<=maxn;j++){
			v[u]=1;
			if(!(i%zs[j])){
				low[u]=low[i]*zs[j];
				if(low[i]==i) d[u]=d[i]+1;
				else d[u]=d[low[u]]*d[i/low[i]];
				break;
			}
			
			low[u]=zs[j];
			d[u]=d[i]<<1;
			miu[u]=-miu[i];
		}
	}
	
	for(int i=1;i<=maxn;i++) d[i]+=d[i-1];
	for(int i=1;i<=maxn;i++) sq[i]=sq[i-1]+miu[i]*miu[i];
}

inline ll getsq(ll x){
	if(x<=maxn) return sq[x];
	
	ll an=0;
	for(int i=1;i*(ll)i<=x;i++){
		an+=miu[i]*(x/(i*(ll)i));
	}
	return an;
}

inline ll getd(ll x){
	if(x<=maxn) return d[x];
	
	ll an=0;
	for(ll i=1,j,now;i<=x;i=j+1){
		now=x/i,j=x/now;
		an+=(j-i+1)*now;
	}
	return an;
}

inline ll query(ll x){
	ll an=0;
	for(ll i=1,j,now;i<=x;i=j+1){
		now=x/i,j=x/now;
		an+=(getsq(j)-getsq(i-1))*getd(now);
	}
	return an;
}

int main(){
	scanf("%d",&T);
	if(T>800) maxn=1000000;
	else maxn=50000000;
	init();
	while(T--){
		scanf("%lld",&n);
		printf("%lld\n",query(n));
	}
	return 0;
}