CF1850F.We Were Both Children

Mihai and Slavic were looking at a group of $n$ frogs, numbered from $1$ to $n$ , all initially located at point $0$ . Frog $i$ has a hop length of $a_i$ .

Each second, frog $i$ hops $a_i$ units forward. Before any frogs start hopping, Slavic and Mihai can place exactly one trap in a coordinate in order to catch all frogs that will ever pass through the corresponding coordinate.

However, the children can't go far away from their home so they can only place a trap in the first $n$ points (that is, in a point with a coordinate between $1$ and $n$ ) and the children can't place a trap in point $0$ since they are scared of frogs.

Can you help Slavic and Mihai find out what is the maximum number of frogs they can catch using a trap?

The first line of the input contains a single integer $t$ ( $1 \le t \le 100$ ) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $n$ ( $1 \leq n \leq 2 \cdot 10^5$ ) — the number of frogs, which equals the distance Slavic and Mihai can travel to place a trap.

The second line of each test case contains $n$ integers $a_1, \ldots, a_n$ ( $1 \leq a_i \leq 10^9$ ) — the lengths of the hops of the corresponding frogs.

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 maximum number of frogs Slavic and Mihai can catch using a trap.

In the first test case, the frogs will hop as follows:

• Frog 1: $0 \to 1 \to 2 \to 3 \to \mathbf{\color{red}{4}} \to \cdots$
• Frog 2: $0 \to 2 \to \mathbf{\color{red}{4}} \to 6 \to 8 \to \cdots$
• Frog 3: $0 \to 3 \to 6 \to 9 \to 12 \to \cdots$
• Frog 4: $0 \to \mathbf{\color{red}{4}} \to 8 \to 12 \to 16 \to \cdots$
• Frog 5: $0 \to 5 \to 10 \to 15 \to 20 \to \cdots$

Therefore, if Slavic and Mihai put a trap at coordinate $4$ , they can catch three frogs: frogs 1, 2, and 4. It can be proven that they can't catch any more frogs.In the second test case, Slavic and Mihai can put a trap at coordinate $2$ and catch all three frogs instantly.