CF1822G2.Magic Triples (Hard Version)

This is the hard version of the problem. The only difference is that in this version, $a_i \le 10^9$ .

For a given sequence of $n$ integers $a$ , a triple $(i, j, k)$ is called magic if:

• $1 \le i, j, k \le n$ .
• $i$ , $j$ , $k$ are pairwise distinct.
• there exists a positive integer $b$ such that $a_i \cdot b = a_j$ and $a_j \cdot b = a_k$ .

Kolya received a sequence of integers $a$ as a gift and now wants to count the number of magic triples for it. Help him with this task!

Note that there are no constraints on the order of integers $i$ , $j$ and $k$ .

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

The first line of the test case contains a single integer $n$ ( $3 \le n \le 2 \cdot 10^5$ ) — the length of the sequence.

The second line of the test contains $n$ integers $a_1, a_2, a_3, \dots, a_n$ ( $1 \le a_i \le 10^9$ ) — the elements of the sequence $a$ .

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 magic triples for the sequence $a$ .

In the first example, there are $6$ magic triples for the sequence $a$ — $(2, 3, 5)$ , $(2, 5, 3)$ , $(3, 2, 5)$ , $(3, 5, 2)$ , $(5, 2, 3)$ , $(5, 3, 2)$ .

In the second example, there is a single magic triple for the sequence $a$ — $(2, 1, 3)$ .