E2. LCM Sum (hard version)
time limit per test
3.5 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output
We are sum for we are many
Some Number

This version of the problem differs from the previous one only in the constraint on $$$t$$$. You can make hacks only if both versions of the problem are solved.

You are given two positive integers $$$l$$$ and $$$r$$$.

Count the number of distinct triplets of integers $$$(i, j, k)$$$ such that $$$l \le i < j < k \le r$$$ and $$$\operatorname{lcm}(i,j,k) \ge i + j + k$$$.

Here $$$\operatorname{lcm}(i, j, k)$$$ denotes the least common multiple (LCM) of integers $$$i$$$, $$$j$$$, and $$$k$$$.

Input

Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$\bf{1 \le t \le 10^5}$$$). Description of the test cases follows.

The only line for each test case contains two integers $$$l$$$ and $$$r$$$ ($$$1 \le l \le r \le 2 \cdot 10^5$$$, $$$l + 2 \le r$$$).

Output

For each test case print one integer — the number of suitable triplets.

Example
Input
5
1 4
3 5
8 86
68 86
6 86868
Output
3
1
78975
969
109229059713337
Note

In the first test case, there are $$$3$$$ suitable triplets:

  • $$$(1,2,3)$$$,
  • $$$(1,3,4)$$$,
  • $$$(2,3,4)$$$.

In the second test case, there is $$$1$$$ suitable triplet:

  • $$$(3,4,5)$$$.