201344: [AtCoder]ARC134 E - Modulo Nim

Problem Statement

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:

$N$
$a_1$ $a_2$ $\cdots$ $a_N$

Output

Print the number of ways to write integers that will result in Taro The First's win if both players play optimally, modulo $998244353$.

Sample Input 1

1
3

Sample 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 $0$.

Sample Input 2

2
5 10

Sample Output 2

47

Sample Input 3

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

Sample Output 3

273710435

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