CF1609C Complex Market Analysis

While performing complex market analysis William encountered the following problem: For a given array $ a $ of size $ n $ and a natural number $ e $ , calculate the number of pairs of natural numbers $ (i, k) $ which satisfy the following conditions: - $ 1 \le i, k $ - $ i + e \cdot k \le n $ . - Product $ a_i \cdot a_{i + e} \cdot a_{i + 2 \cdot e} \cdot \ldots \cdot a_{i + k \cdot e} $ is a prime number. A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10\,000 $ ). Description of the test cases follows. The first line of each test case contains two integers $ n $ and $ e $ $ (1 \le e \le n \le 2 \cdot 10^5) $ , the number of items in the array and number $ e $ , respectively. The second line contains $ n $ integers $ a_1, a_2, \dots, a_n $ $ (1 \le a_i \le 10^6) $ , the contents of the array. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case output the answer in the following format: Output one line containing the number of pairs of numbers $ (i, k) $ which satisfy the conditions.

In the first example test case two pairs satisfy the conditions: 1. $ i = 2, k = 1 $ , for which the product is: $ a_{2} \cdot a_{5} = 2 $ which is a prime number. 2. $ i = 3, k = 1 $ , for which the product is: $ a_{3} \cdot a_{6} = 19 $ which is a prime number. In the second example test case there are no pairs that satisfy the conditions. In the third example test case four pairs satisfy the conditions: 1. $ i = 1, k = 1 $ , for which the product is: $ a_{1} \cdot a_{4} = 2 $ which is a prime number. 2. $ i = 1, k = 2 $ , for which the product is: $ a_{1} \cdot a_{4} \cdot a_{7} = 2 $ which is a prime number. 3. $ i = 3, k = 1 $ , for which the product is: $ a_{3} \cdot a_{6} = 2 $ which is a prime number. 4. $ i = 6, k = 1 $ , for which the product is: $ a_{6} \cdot a_{9} = 2 $ which is a prime number. In the fourth example test case there are no pairs that satisfy the conditions. In the fifth example test case five pairs satisfy the conditions: 1. $ i = 1, k = 1 $ , for which the product is: $ a_{1} \cdot a_{2} = 2 $ which is a prime number. 2. $ i = 1, k = 2 $ , for which the product is: $ a_{1} \cdot a_{2} \cdot a_{3} = 2 $ which is a prime number. 3. $ i = 1, k = 3 $ , for which the product is: $ a_{1} \cdot a_{2} \cdot a_{3} \cdot a_{4} = 2 $ which is a prime number. 4. $ i = 2, k = 1 $ , for which the product is: $ a_{2} \cdot a_{3} = 2 $ which is a prime number. 5. $ i = 2, k = 2 $ , for which the product is: $ a_{2} \cdot a_{3} \cdot a_{4} = 2 $ which is a prime number. In the sixth example test case there are no pairs that satisfy the conditions.