A16381 | Partitioning the Array

Allen has an array $a_1, a_2,\ldots,a_n$ . For every positive integer $k$ that is a divisor of $n$ , Allen does the following:

- He partitions the array into $\frac{n}{k}$ disjoint subarrays of length $k$ . In other words, he partitions the array into the following subarrays: $$$$[a_1,a_2,\ldots,a_k],[a_{k+1}, a_{k+2},\ldots,a_{2k}],\ldots,[a_{n-k+1},a_{n-k+2},\ldots,a_{n}] $$ </li><li> Allen earns one point if there exists some positive integer $m$ ( $m \\geq 2$ ) such that if he replaces every element in the array with its remainder when divided by $m$$$, then all subarrays will be identical.

Help Allen find the number of points he will earn.

Each test consists of multiple test cases. The first line contains a single integer $t$ ( $1 \leq t \leq 10^4$ ) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $1 \leq n \leq 2\cdot10^5$ ) — the length of the array $a$ .

The second line of each test case contains $n$ integers $a_1, a_2,\ldots, a_n$ ( $1 \leq a_i \leq n$ ) — the elements of the array $a$ .

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, output a single integer — the number of points Allen will earn.