Preparando MOJI
For two positive integers $$$l$$$ and $$$r$$$ ($$$l \le r$$$) let $$$c(l, r)$$$ denote the number of integer pairs $$$(i, j)$$$ such that $$$l \le i \le j \le r$$$ and $$$\operatorname{gcd}(i, j) \ge l$$$. Here, $$$\operatorname{gcd}(i, j)$$$ is the greatest common divisor (GCD) of integers $$$i$$$ and $$$j$$$.
YouKn0wWho has two integers $$$n$$$ and $$$k$$$ where $$$1 \le k \le n$$$. Let $$$f(n, k)$$$ denote the minimum of $$$\sum\limits_{i=1}^{k}{c(x_i+1,x_{i+1})}$$$ over all integer sequences $$$0=x_1 \lt x_2 \lt \ldots \lt x_{k} \lt x_{k+1}=n$$$.
Help YouKn0wWho find $$$f(n, k)$$$.
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 3 \cdot 10^5$$$) — the number of test cases.
The first and only line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k \le n \le 10^5$$$).
For each test case, print a single integer — $$$f(n, k)$$$.
4 6 2 4 4 3 1 10 3
8 4 6 11
In the first test case, YouKn0wWho can select the sequence $$$[0, 2, 6]$$$. So $$$f(6, 2) = c(1, 2) + c(3, 6) = 3 + 5 = 8$$$ which is the minimum possible.
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Provedor Codeforces
Código CF1603D
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Datas 09/05/2023 10:19:25
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