Modulo Nim

AtCoder

IDarc134_e

Time6000ms

Memory256MB

Difficulty—

Snuke found a blackboard with nothing written on it. He will do $N$ operations on this blackboard. In the $i$\-th operation, he chooses an integer between $1$ and $a_i$ (inclusive) and writes it on the blackboard. After $N$ integers are written on the blackboard, Taro The First and Jiro The Second will play a game using it. In the game, the two alternately do the operation below, with Taro The First going first. * Let $X$ be the largest integer written on the blackboard. * If $X=0$, the current player **wins** and the game ends. * Choose an integer $m$ between $1$ and $X$ (inclusive). * Replace each of the $N$ integers written on the blackboard with its remainder when divided by $m$. There are $\prod_{i=1}^{N}a_i$ ways for Snuke to write numbers on the blackboard. Find the number of ways among them that will result in Taro The First's win if both players play optimally, modulo $998244353$. ## Constraints * All values in input are integers. * $1 \leq N \leq 200$ * $1 \leq a_i \leq 200$ ## Input Input is given from Standard Input in the following format: $N$ $a_1$ $a_2$ $\cdots$ $a_N$ [samples]

Samples

Input #1

1
3

Output #1

1

*   Taro The First will win only if Snuke writes $3$.
*   Otherwise, Taro The First can only play a move that makes the integer on the blackboard $0$.

Input #2

2
5 10

Output #2

Input #3

20
200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200

Output #3

273710435

*   Be sure to find the count modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "Modulo Nim",
    "description": {
      "content": "Snuke found a blackboard with nothing written on it. He will do $N$ operations on this blackboard. In the $i$\\-th operation, he chooses an integer between $1$ and $a_i$ (inclusive) and writes it on th",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 6000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc134_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Snuke found a blackboard with nothing written on it.\nHe will do $N$ operations on this blackboard. In the $i$\\-th operation, he chooses an integer between $1$ and $a_i$ (inclusive) and writes it on th...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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