A Perfect Problem

2000ms

262144K

Description:

A sequence of integers $$$b_1, b_2, \ldots, b_m$$$ is called good if $$$max(b_1, b_2, \ldots, b_m) \cdot min(b_1, b_2, \ldots, b_m) \ge b_1 + b_2 + \ldots + b_m$$$.

A sequence of integers $$$a_1, a_2, \ldots, a_n$$$ is called perfect if every non-empty subsequence of $$$a$$$ is good.

YouKn0wWho has two integers $$$n$$$ and $$$M$$$, $$$M$$$ is prime. Help him find the number, modulo $$$M$$$, of perfect sequences $$$a_1, a_2, \ldots, a_n$$$ such that $$$1 \le a_i \le n + 1$$$ for each integer $$$i$$$ from $$$1$$$ to $$$n$$$.

A sequence $$$d$$$ is a subsequence of a sequence $$$c$$$ if $$$d$$$ can be obtained from $$$c$$$ by deletion of several (possibly, zero or all) elements.

Input:

The first and only line of the input contains two space-separated integers $$$n$$$ and $$$M$$$ ($$$1 \le n \le 200$$$; $$$10^8 \le M \le 10^9$$$). It is guaranteed that $$$M$$$ is prime.

Output:

Print a single integer — the number of perfect sequences modulo $$$M$$$.

Sample Input:

2 998244353

Sample Output:

Sample Input:

4 100000007

Sample Output:

Sample Input:

69 999999937

Sample Output:

456886663

Note:

In the first test case, the perfect sequences are $$$[2, 2]$$$, $$$[2, 3]$$$, $$$[3, 2]$$$ and $$$[3, 3]$$$.

In the second test case, some of the perfect sequences are $$$[3, 4, 3, 5]$$$, $$$[4, 5, 4, 4]$$$, $$$[4, 5, 5, 5]$$$ etc. One example of a sequence which is not perfect is $$$[2, 3, 3, 4]$$$, because, for example, the subsequence $$$[2, 3, 4]$$$ is not an good as $$$2 \cdot 4 < 2 + 3 + 4$$$.

Informação

Provedor Codeforces

Código CF1603E