306294: CF1174E. Ehab and the Expected GCD Problem
Memory Limit:256 MB
Time Limit:2 S
Judge Style:Text Compare
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Description
Ehab and the Expected GCD Problem
题意翻译
$p$ 是一个排列,定义 $f(p)$ : 设 $g_i$ 为 $p_1, p_2, \cdots, p_i$ 的最大公因数(即前缀最大公因数),则 $f(p)$ 为 $g_1,g_2,\cdots,g_n$ 中不同的数的个数。 设 $f_{max}(n)$ 为 $1,2,\cdots,n$ 的所有排列 $p$ 中 $f(p)$ 的最大值,你需要求有多少 $1,2,\cdots,n$ 的排列 $p$ 满足 $f(p)=f_{max}(n)$,答案对 $10^9+7$ 取模。题目描述
Let's define a function $ f(p) $ on a permutation $ p $ as follows. Let $ g_i $ be the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of elements $ p_1 $ , $ p_2 $ , ..., $ p_i $ (in other words, it is the GCD of the prefix of length $ i $ ). Then $ f(p) $ is the number of distinct elements among $ g_1 $ , $ g_2 $ , ..., $ g_n $ . Let $ f_{max}(n) $ be the maximum value of $ f(p) $ among all permutations $ p $ of integers $ 1 $ , $ 2 $ , ..., $ n $ . Given an integers $ n $ , count the number of permutations $ p $ of integers $ 1 $ , $ 2 $ , ..., $ n $ , such that $ f(p) $ is equal to $ f_{max}(n) $ . Since the answer may be large, print the remainder of its division by $ 1000\,000\,007 = 10^9 + 7 $ .输入输出格式
输入格式
The only line contains the integer $ n $ ( $ 2 \le n \le 10^6 $ ) — the length of the permutations.
输出格式
The only line should contain your answer modulo $ 10^9+7 $ .
输入输出样例
输入样例 #1
2
输出样例 #1
1
输入样例 #2
3
输出样例 #2
4
输入样例 #3
6
输出样例 #3
120