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.

In the first test case, $k=2$ earns a point since Allen can pick $m = 2$ and both subarrays will be equal to $[1, 0]$ . $k=4$ also earns a point, since no matter what $m$ Allen chooses, there will be only one subarray and thus all subarrays are equal.

In the second test case, Allen earns $1$ point for $k=3$ , where his choice for $m$ does not matter.