CF2159D2 Inverse Minimum Partition (Hard Version)

This is the hard version of the problem. The difference between the versions is that in this version, you are asked to find the sum of $ f(b) $ over all contiguous subsequences $ b $ of $ a $ . You can hack only if you solved all versions of this problem. For some sequence $ b $ of $ k $ positive integers, the cost of the sequence is defined as follows $ ^{\text{∗}} $ : $$$ \mathtt{cost}(b)=\left\lceil{\frac{b_k}{\min(b_1,b_2,\ldots,b_k)}}\right\rceil $$$ Assume that you partition a sequence $ c $ into one or more sequences, so that the sequences become $ c $ when concatenated. For example, when the sequence $ c $ is $ [3,1,4,1,5] $ , a few valid ways to partition the sequence are $ [[3,1],[4,1,5]] $ , $ [[3],[1,4,1],[5]] $ , $ [[3,1,4,1,5]] $ . On the other hand, $ [[3,1],[1,5]] $ and $ [[3,4,1],[1,5]] $ are not valid ways to partition the sequence. For a partition of a sequence $ c $ of positive integers, the total cost of the partition is defined as the sum of the costs of each sequence in the partition. Then, let us define $ f(c) $ as the minimum total cost to partition the sequence $ c $ . Given a sequence $ a $ of $ n $ positive integers, you must compute the following: $$$ \sum_{l=1}^{n} {\sum_{r=l}^{n} f([a_l,a_{l+1},\ldots,a_r])} $$$ $ ^{\text{∗}} $ Given a real value $ x $ , $ \left\lceil{x}\right\rceil $ is defined as the smallest integer no less than $ x $ . For example, the value of $ \left\lceil{3.14}\right\rceil $ is $ 4 $ .

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^4 $ ). The description of the test cases follows. The first line of each test case contains a single integer $ n $ ( $ 1 \le n \le 4 \cdot 10^5 $ ). The second line of each test case contains $ a_1,a_2,\ldots,a_n $ ( $ 1 \le a_i \le 10^{18} $ ). It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 4 \cdot 10^5 $ .

For each test case, output the answer to the problem on a separate line.

For the first test case, the following are all contiguous subsequences of $ a $ and their corresponding values: - $ [3] \to [[3]] $ : $ f([3])=1 $ ; - $ [3,1] \to [[3,1]] $ : $ f([3,1])=1 $ ; - $ [3,1,4] \to [[3,1],[4]] $ : $ f([3,1,4])=2 $ ; - $ [3,1,4,1] \to [[3,1,4,1]] $ : $ f([3,1,4,1])=1 $ ; - $ [3,1,4,1,5] \to [[3,1,4,1],[5]] $ : $ f([3,1,4,1,5])=2 $ ; - $ [1] \to [[1]] $ : $ f([1])=1 $ ; - $ [1,4] \to [[1],[4]] $ : $ f([1,4])=2 $ ; - $ [1,4,1] \to [[1,4,1]] $ : $ f([1,4,1])=1 $ ; - $ [1,4,1,5] \to [[1,4,1],[5]] $ : $ f([1,4,1,5])=2 $ ; - $ [4] \to [[4]] $ : $ f([4])=1 $ ; - $ [4,1] \to [[4,1]] $ : $ f([4,1])=1 $ ; - $ [4,1,5] \to [[4,1],[5]] $ : $ f([4,1,5])=2 $ ; - $ [1] \to [[1]] $ : $ f([1])=1 $ ; - $ [1,5] \to [[1],[5]] $ : $ f([1,5])=2 $ ; - $ [5] \to [[5]] $ : $ f([5])=1 $ . Therefore, the answer for the first test case is $ 1+1+2+1+2+1+2+1+2+1+1+2+1+2+1=21 $ .