codeforces 776E - The Holmes Children

最新推荐文章于 2018-04-27 23:06:21 发布

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最新推荐文章于 2018-04-27 23:06:21 发布

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分类专栏： Codeforces 文章标签： codeforces 欧拉函数

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本文介绍了一个复杂的数学问题，该问题是关于寻找特定数值的过程。通过递归定义的k-composite函数Fk(n)，并求解该函数的模1000000007值。文中提供了一种算法实现，利用数学性质简化问题，最终得出解决方案。

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Problem Description
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 . 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:

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.

Example
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 and gcd(x, y) = 1. Hence, f(7) = 6. So, F1(7) = f(g(7)) = f(f(7) + f(1)) = f(6 + 1) = f(7) = 6.

g(n)：f(x)的和，n为x的倍数。eg：g(10)=f(1)+f(2)+f(5)
一系列代入发现g(n)=n;

将g(n)=n代入Fk(n),得Fk(n)=f(f(f…..f(n)…)其中嵌套的f为（k+1）/2层，
eg：F5(n)=f(f(f(n)))；

f(n)：任意的两个数x,y,满足x+y=n且x,y互为质数
假设如果x和y不互质，那么设，x和y的共同因子为w，又因为n=x+y，所以可以提出公因子w，所以n和x，y都不互质；相反，如果x,y互质，则n与他们中的任意一个互质。所以只找满足与n互质的x 数量就可以了。
互质数的求法：http://blog.youkuaiyun.com/winter2121/article/details/78411647

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<math.h>
using namespace std;
typedef long long ll;
ll print(ll n){
    ll m=sqrt(n);
    ll x=n;
    for(ll i=2;i<=m&&i<=n;++i){
        if(n%i==0) x=x/i*(i-1);
        while(n%i==0){
            n/=i;
        }

    }
    if(n>1)
        x=x/n*(n-1);
    return x;
}
int main()
{
    ll n,k;
    cin>>n>>k;
    k=(k+1)/2;
    while(k--&&n>1){
        n=print(n);
    }
    cout<<n%1000000007<<endl;
    return 0;
}