Makoto and a Blackboard

题意翻译

给定 $n,k$，一共会进行 $k$ 次操作，每次操作会把 $n$ 等概率的变成 $n$ 的某个约数 求操作 $k$ 次后 $n$ 的期望是多少，答案对 $10^9+7$ 取模 $1 \le n \le 10^{15},1 \le k \le 10^4$

题目描述

Makoto has a big blackboard with a positive integer $ n $ written on it. He will perform the following action exactly $ k $ times: Suppose the number currently written on the blackboard is $ v $ . He will randomly pick one of the divisors of $ v $ (possibly $ 1 $ and $ v $ ) and replace $ v $ with this divisor. As Makoto uses his famous random number generator (RNG) and as he always uses $ 58 $ as his generator seed, each divisor is guaranteed to be chosen with equal probability. He now wonders what is the expected value of the number written on the blackboard after $ k $ steps. It can be shown that this value can be represented as $ \frac{P}{Q} $ where $ P $ and $ Q $ are coprime integers and $ Q \not\equiv 0 \pmod{10^9+7} $ . Print the value of $ P \cdot Q^{-1} $ modulo $ 10^9+7 $ .

输入输出格式

输入格式

The only line of the input contains two integers $ n $ and $ k $ ( $ 1 \leq n \leq 10^{15} $ , $ 1 \leq k \leq 10^4 $ ).

输出格式

Print a single integer — the expected value of the number on the blackboard after $ k $ steps as $ P \cdot Q^{-1} \pmod{10^9+7} $ for $ P $ , $ Q $ defined above.

输入输出样例

输入样例 #1

6 1

输出样例 #1

3

输入样例 #2

6 2

输出样例 #2

875000008

输入样例 #3

60 5

输出样例 #3

237178099

说明

In the first example, after one step, the number written on the blackboard is $ 1 $ , $ 2 $ , $ 3 $ or $ 6 $ — each occurring with equal probability. Hence, the answer is $ \frac{1+2+3+6}{4}=3 $ . In the second example, the answer is equal to $ 1 \cdot \frac{9}{16}+2 \cdot \frac{3}{16}+3 \cdot \frac{3}{16}+6 \cdot \frac{1}{16}=\frac{15}{8} $ .

题意翻译

给定 $n,k$，一共会进行 $k$ 次操作，每次操作会把 $n$ 等概率的变成 $n$ 的某个约数 求操作 $k$ 次后 $n$ 的期望是多少，答案对 $10^9+7$ 取模 $1 \le n \le 10^{15},1 \le k \le 10^4$