Score : $600$ points

问题描述

给定正整数 $N$ 和 $M$，其中保证 $N\leq M \leq 2N$。

计算满足以下条件的所有正整数序列 $S=(S_1,S_2,\dots,S_N)$ 的下述值之和，并对结果取模 $200\ 003$（一个质数）：

• $\displaystyle \prod_{k=1}^{N} \min(k,S_k)$ （注意此处的特殊模运算）。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given positive integers $N$ and $M$. Here, it is guaranteed that $N\leq M \leq 2N$.

Print the sum, modulo $200\ 003$ (a prime), of the following value over all sequences of positive integers $S=(S_1,S_2,\dots,S_N)$ such that $\displaystyle \sum_{i=1}^{N} S_i = M$ (notice the unusual modulo):

• $\displaystyle \prod_{k=1}^{N} \min(k,S_k)$.

Constraints

• $1 \leq N \leq 10^{12}$
• $N \leq M \leq 2N$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$

Output

Print the answer as an integer.

Sample Input 1

3 5

Sample Output 1

14

There are six sequences $S$ that satisfy the condition: $S=(1,1,3), S=(1,2,2), S=(1,3,1), S=(2,1,2), S=(2,2,1), S=(3,1,1)$.

The value $\displaystyle \prod_{k=1}^{N} \min(k,S_k)$ for each of those $S$ is as follows.

• $S=(1,1,3)$ : $1\times 1 \times 3 = 3$
• $S=(1,2,2)$ : $1\times 2 \times 2 = 4$
• $S=(1,3,1)$ : $1\times 2 \times 1 = 2$
• $S=(2,1,2)$ : $1\times 1 \times 2 = 2$
• $S=(2,2,1)$ : $1\times 2 \times 1 = 2$
• $S=(3,1,1)$ : $1\times 1 \times 1 = 1$

Thus, you should print their sum: $14$.

Sample Input 2

1126 2022

Sample Output 2

40166

Print the sum modulo $200\ 003$.

Sample Input 3

1000000000000 1500000000000

Sample Output 3

180030

update @ 2024/3/10 11:45:33