CF1174E Ehab and the Expected GCD Problem

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 $ .

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Consider the second example: these are the permutations of length $ 3 $ : - $ [1,2,3] $ , $ f(p)=1 $ . - $ [1,3,2] $ , $ f(p)=1 $ . - $ [2,1,3] $ , $ f(p)=2 $ . - $ [2,3,1] $ , $ f(p)=2 $ . - $ [3,1,2] $ , $ f(p)=2 $ . - $ [3,2,1] $ , $ f(p)=2 $ . The maximum value $ f_{max}(3) = 2 $ , and there are $ 4 $ permutations $ p $ such that $ f(p)=2 $ .