Codeforces 776E 数论

最新推荐文章于 2024-04-25 07:22:11 发布

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最新推荐文章于 2024-04-25 07:22:11 发布

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本文探讨了Eurus提出的复合函数Fk(n)的问题，这是一个递归定义的函数，涉及欧拉函数φ(n)的计算。文章提供了一个有效求解Fk(n) mod 1000000007的算法实现。

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The Holmes children are fighting over who amongst them is the cleverest.

Mycroft asked Sherlock and Eurus to find value of f(n), where f(1) = 1 and for n ≥ 2, f(n) is the number of distinct ordered positive integer pairs (x, y) that satisfy x + y = n and gcd(x, y) = 1. The integer gcd(a, b) is the greatest common divisor of a and b.

Sherlock said that solving this was child's play and asked Mycroft to instead get the value of $\text{[math]}$ . Summation is done over all positive integers d that divide n.

Eurus was quietly observing all this and finally came up with her problem to astonish both Sherlock and Mycroft.

She defined a k-composite function Fk(n) recursively as follows:

$\text{[math]}$

She wants them to tell the value of Fk(n) modulo 1000000007.

Input

A single line of input contains two space separated integers n (1 ≤ n ≤ 1012) and k (1 ≤ k ≤ 1012) indicating that Eurus asks Sherlock and Mycroft to find the value of Fk(n) modulo 1000000007.

Output

Output a single integer — the value of Fk(n) modulo 1000000007.

Examples

input
7 1

output
6

input
10 2

output
4

Note

In the first case, there are 6 distinct ordered pairs (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1) satisfying x + y = 7 andgcd(x, y) = 1. Hence, f(7) = 6. So, F1(7) = f(g(7)) = f(f(7) + f(1)) = f(6 + 1) = f(7) = 6.

题意：求fk(n)%1000000007。

题解：

f(n)为欧拉函数

g(n)恒等于n

推荐看下大牛的证明QAQ

因为每走两步n都会取一次phi，所以可以直接暴力。

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
typedef long long ll;
ll phi(ll n){  
    ll ans=n,i;  
    for(i=2;i*i<=n;i++){  
        if(n%i==0){  
            ans=ans-ans/i;  
            while(n%i==0)n/=i;  
        }  
    }  
    if(n>1)ans=ans-ans/n;  
    return ans;  
}  
int main(){
	ll n,k;
	cin>>n>>k;
	while(k>0&&n>1){
		if(k%2==1)n=phi(n);
		k--;
	}
	cout<<n%1000000007<<endl;
	return 0;
}