CF2171H Shiori Miyagi and Maximum Array Score

"Are you stupid, Sendai-san?" — Shiori Miyagi For the cost of 5000 yen, Miyagi can ask Sendai to do whatever she wants! Today, Miyagi demands that Sendai make her an array... specifically, Miyagi only wants a very particular type of array. For arbitrary integers $ b\geq 2 $ and $ x\geq 1 $ , define $ v(b, x) $ to be the maximal $ k $ satisfying $ b^k \mid x $ ; that is, the maximal $ k $ such that $ x $ is a multiple of $ b^k $ . It can be shown that this is always a well-defined, nonnegative integer. You are given integers $ n $ and $ m $ satisfying $ n\leq m $ . Find the maximum value of $ \sum_{i=2}^n v(i, a_i) $ across all arrays $ a $ of length $ n $ satisfying the following conditions: - $ a $ is strictly increasing; that is, for all $ 1\leq i\leq n-1 $ , $ a_i < a_{i+1} $ , and - for all $ 1\leq i\leq n $ , $ 1\leq a_i\leq m $ .

The first line contains a single integer $ t $ ( $ 1 \leq t \leq 10^4 $ ) — the number of test cases. The only line of each test case contains two integers $ n $ and $ m $ ( $ 2\leq n\leq m \leq 2\cdot 10^5 $ ). It is guaranteed that the sum of $ m $ over all test cases does not exceed $ 2\cdot 10^5 $ .

For each test case, output a single integer, the maximum value of $ \sum_{i=2}^n v(i, a_i) $ across all arrays $ a $ of length $ n $ satisfying the given conditions.

In the first example, one possible array is $ a = [6, 8, 9, 16] $ which yields a value of $ 3 + 2 + 2 = 7 $ . It can be shown that this is the maximum value of $ \sum_{i=2}^n v(i, a_i) $ across all arrays $ a $ of length $ n $ satisfying the given conditions.