Fraction Line

AtCoder

IDarc192_d

Time2000ms

Memory256MB

Difficulty—

For a positive rational number $x$, define $f(x)$ as follows: > Express $x$ as $\dfrac{P}{Q}$ using coprime positive integers $P$ and $Q$. $f(x)$ is defined as the value $P\times Q$. You are given a positive integer $N$ and a sequence $A=(A_1,A_2,\dots,A_{N-1})$ of positive integers of length $N-1$. We call a sequence $S=(S_1,S_2,\dots,S_N)$ of positive integers of length $N$ a **good sequence** if it satisfies all of the following conditions: * For every integer $i$ with $1\leq i\leq N-1$, it holds that $f\left(\dfrac{S_i}{S_{i+1}}\right)=A_i$. * $\gcd(S_1,S_2,\dots,S_N)=1$. Define the **score** of a sequence as the product of all its elements. It can be proved that there are finitely many good sequences. Find the sum, modulo $998244353$, of the scores of all good sequences. ## Constraints * $2\leq N\leq 1000$ * $1\leq A_i\leq 1000$ $(1\leq i\leq N-1)$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\dots$ $A_{N-1}$ [samples]

Samples

Input #1

6
1 9 2 2 9

Output #1

939634344

For example, both $(2,2,18,9,18,2)$ and $(18,18,2,1,2,18)$ are good sequences, and both have a score of $23328$.
There are a total of $16$ good sequences, and the sum of the scores of all of them is $939634344$.

Input #2

2
9

Output #2

18

There are $2$ good sequences, both with a score of $9$.

Input #3

25
222 299 229 22 999 922 99 992 22 292 222 229 992 922 22 992 222 222 99 29 92 999 2 29

Output #3

192457116

Do not forget to compute the sum modulo $998244353$.

API Response (JSON)

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    "name": "Fraction Line",
    "description": {
      "content": "For a positive rational number $x$, define $f(x)$ as follows: > Express $x$ as $\\dfrac{P}{Q}$ using coprime positive integers $P$ and $Q$. $f(x)$ is defined as the value $P\\times Q$. You are given a",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc192_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For a positive rational number $x$, define $f(x)$ as follows:\n\n> Express $x$ as $\\dfrac{P}{Q}$ using coprime positive integers $P$ and $Q$. $f(x)$ is defined as the value $P\\times Q$.\n\nYou are given a...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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