#P1822G2. Magic Triples (Hard Version)
Magic Triples (Hard Version)
Description
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 $a$.
Input
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 $2 \cdot 10^5$.
Output
For each test case, output a single integer — the number of magic triples for the sequence $a$.
7
5
1 7 7 2 7
3
6 2 18
9
1 2 3 4 5 6 7 8 9
4
1000 993 986 179
7
1 10 100 1000 10000 100000 1000000
8
1 1 2 2 4 4 8 8
9
1 1 1 2 2 2 4 4 4
6
1
3
0
9
16
45
Note
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 $a$ — $(2, 1, 3)$.