Description:
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.
Input Format:
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 $$$2 \cdot 10^5$$$.
Output Format:
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.
Note:
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.