codeforces#P1712E2. LCM Sum (hard version)
LCM Sum (hard version)
Description
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 $k$.
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 $r$ ($1 \le l \le r \le 2 \cdot 10^5$, $l + 2 \le r$).
For each test case print one integer — the number of suitable triplets.
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.
Samples
<div class="test-example-line test-example-line-even test-example-line-0">5</div><div class="test-example-line test-example-line-odd test-example-line-1">1 4</div><div class="test-example-line test-example-line-even test-example-line-2">3 5</div><div class="test-example-line test-example-line-odd test-example-line-3">8 86</div><div class="test-example-line test-example-line-even test-example-line-4">68 86</div><div class="test-example-line test-example-line-odd test-example-line-5">6 86868</div><div class="test-example-line test-example-line-odd test-example-line-5"></div>
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:
- $(3,4,5)$.