Score : $500$ points

问题描述

已知一个包含 $N$ 个数的序列 $A$。找出将 $A$ 分割成若干个非空连续子序列 $B_1, B_2, \ldots, B_k$ 的方法数量，使得满足以下条件：

• 对于每个 $i\ (1 \leq i \leq k)$，子序列 $B_i$ 中元素的和能够被 $i$ 整除。

由于计数结果可能会非常大，请以 $(10^9+7)$ 为模，输出最终答案。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Given is a sequence $A$ of $N$ numbers. Find the number of ways to separate $A$ into some number of non-empty contiguous subsequence $B_1, B_2, \ldots, B_k$ so that the following condition is satisfied:

• For every $i\ (1 \leq i \leq k)$, the sum of elements in $B_i$ is divisible by $i$.

Since the count can be enormous, print it modulo $(10^9+7)$.

Constraints

• $1 \leq N \leq 3000$
• $1 \leq A_i \leq 10^{15}$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the number of ways to separate the sequence so that the condition in the Problem Statement is satisfied, modulo $(10^9+7)$.

Sample Input 1

4
1 2 3 4

Sample Output 1

3

We have three ways to separate the sequence, as follows:

• $(1),(2),(3),(4)$
• $(1,2,3),(4)$
• $(1,2,3,4)$

Sample Input 2

5
8 6 3 3 3

Sample Output 2

5

Sample Input 3

10
791754273866483 706434917156797 714489398264550 918142301070506 559125109706263 694445720452148 648739025948445 869006293795825 718343486637033 934236559762733

Sample Output 3

15

The values in input may not fit into a $32$-bit integer type.

update @ 2024/3/10 09:21:03