Sum of Mod

AtCoder

IDarc208_b

Time2000ms

Memory256MB

Difficulty—

You are given positive integers $N$ and $K$. Find one length-$N$ sequence of positive integers $A=(A_1,A_2,\ldots,A_N)$ with the minimum value of $A_N$ among the ones that satisfy all of the following conditions. * $A$ is non-decreasing. That is, $A_i \le A_{i+1}$ for $1\le i\le N-1$. * $\displaystyle \sum_{i=1}^{N-1} (A_{i+1} \bmod A_i)=K$. You are given $T$ test cases; solve each of them. ## Constraints * $1\le T \le 10^5$ * $2\le N \le 2\times 10^5$ * The sum of $N$ over all test cases is at most $2\times 10^5$. * $1\le K\le 10^9$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $T$ $\text{case}_1$ $\text{case}_2$ $\vdots$ $\text{case}_T$ Each test case is given in the following format: $N$ $K$ [samples]

Samples

Input #1

2
5 3
2 3

Output #1

1 2 3 4 5
4 7

Consider the first test case.
$A=(1,2,3,4,5)$ is non-decreasing and satisfies $\displaystyle \sum_{i=1}^{N-1} (A_{i+1} \bmod A_i) = (2\bmod 1) + (3\bmod 2) + (4\bmod 3) + (5\bmod 4)=3=K$, so it satisfies all conditions.
There does not exist an $A$ that satisfies all conditions and has a value of $A_N$ less than $5$, so printing $A=(1,2,3,4,5)$ will be accepted.
Other than this, printing $A=(2,3,4,5,5)$ or $A=(2,3,3,5,5)$ will also be accepted.

API Response (JSON)

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      "statement_type": "Markdown",
      "content": "You are given positive integers $N$ and $K$.\nFind one length-$N$ sequence of positive integers $A=(A_1,A_2,\\ldots,A_N)$ with the minimum value of $A_N$ among the ones that satisfy all of the following...",
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