codeforces#P1603E. A Perfect Problem
A Perfect Problem
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.
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.
Print a single integer — the number of perfect sequences modulo $M$.
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 $M$.
Samples
2 998244353
4
4 100000007
32
69 999999937
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 $2 \cdot 4 < 2 + 3 + 4$.