#P1967B2. Reverse Card (Hard Version)
Reverse Card (Hard Version)
Description
The two versions are different problems. You may want to read both versions. You can make hacks only if both versions are solved.
You are given two positive integers $n$, $m$.
Calculate the number of ordered pairs $(a, b)$ satisfying the following conditions:
- $1\le a\le n$, $1\le b\le m$;
- $b \cdot \gcd(a,b)$ is a multiple of $a+b$.
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1\le t\le 10^4$). The description of the test cases follows.
The first line of each test case contains two integers $n$, $m$ ($1\le n,m\le 2 \cdot 10^6$).
It is guaranteed that neither the sum of $n$ nor the sum of $m$ over all test cases exceeds $2 \cdot 10^6$.
For each test case, print a single integer: the number of valid pairs.
Input
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1\le t\le 10^4$). The description of the test cases follows.
The first line of each test case contains two integers $n$, $m$ ($1\le n,m\le 2 \cdot 10^6$).
It is guaranteed that neither the sum of $n$ nor the sum of $m$ over all test cases exceeds $2 \cdot 10^6$.
Output
For each test case, print a single integer: the number of valid pairs.
6
1 1
2 3
3 5
10 8
100 1233
1000000 1145141
0
1
1
6
423
5933961
Note
In the first test case, no pair satisfies the conditions.
In the fourth test case, $(2,2),(3,6),(4,4),(6,3),(6,6),(8,8)$ satisfy the conditions.