Sum Avoidance

AtCoder

IDagc057_d

Time5000ms

Memory256MB

Difficulty—

You are given positive integers $S$ and $K$. A sequence of positive integers $A = (A_1, A_2, \ldots, A_N)$ is said to be a **good sequence** when satisfying the following two conditions. * $1\leq A_1 < A_2 < \cdots < A_N \leq S - 1$ holds. * $\sum_{i=1}^NA_ix_i\neq S$ holds for any sequence of non-negative integers $(x_1, x_2, \ldots, x_N)$. Let $A = (A_1, A_2, \ldots, A_N)$ be the lexicographically smallest of the good sequences with the maximum number $N$ of terms. Print the $K$\-th term $A_K$ of this sequence, or `-1` if $K > N$. We will give you $T$ test cases; solve each of them. ## Constraints * $1\leq T\leq 1000$ * $3\leq S\leq 10^{18}$ * $1\leq K \leq S - 1$ ## Input Input is given from Standard Input in the following format: $T$ $\text{case}_1$ $\vdots$ $\text{case}_T$ Each case is in the following format: $S$ $K$ [samples]

Samples

Input #1

13
3 1
3 2
7 1
7 2
7 3
7 4
10 1
10 2
10 3
10 4
10 5
2022 507
1000000000000000000 999999999999999999

Output #1

2
-1
2
4
6
-1
3
6
8
9
-1
1351
-1

For the cases $S = 3, 7, 10$, the sequence $A$ will be as follows.

*   For $S=3$: $A = (2)$.
*   For $S=7$: $A = (2,4,6)$.
*   For $S=10$: $A = (3,6,8,9)$.

API Response (JSON)

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  "problem": {
    "name": "Sum Avoidance",
    "description": {
      "content": "You are given positive integers $S$ and $K$. A sequence of positive integers $A = (A_1, A_2, \\ldots, A_N)$ is said to be a **good sequence** when satisfying the following two conditions. *   $1\\leq A",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc057_d"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positive integers $S$ and $K$. A sequence of positive integers $A = (A_1, A_2, \\ldots, A_N)$ is said to be a **good sequence** when satisfying the following two conditions.\n\n*   $1\\leq A...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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