Array Equalizer
Description
Jeevan has two arrays $a$ and $b$ of size $n$. He is fond of performing weird operations on arrays. This time, he comes up with two types of operations:
- Choose any $i$ ($1 \le i \le n$) and increment $a_j$ by $1$ for every $j$ which is a multiple of $i$ and $1 \le j \le n$.
- Choose any $i$ ($1 \le i \le n$) and decrement $a_j$ by $1$ for every $j$ which is a multiple of $i$ and $1 \le j \le n$.
He wants to convert array $a$ into an array $b$ using the minimum total number of operations. However, Jeevan seems to have forgotten the value of $b_1$. So he makes some guesses. He will ask you $q$ questions corresponding to his $q$ guesses, the $i$-th of which is of the form:
- If $b_1 = x_i$, what is the minimum number of operations required to convert $a$ to $b$?
Help him by answering each question.
The first line contains a single integer $n$ $(1 \le n \le 2 \cdot 10^{5})$ — the size of arrays $a$ and $b$.
The second line contains $n$ integers $a_1, a_2, ..., a_n$ $(1 \le a_i \le 10^6)$.
The third line contains $n$ integers $b_1, b_2, ..., b_n$ $(1 \le b_i \le 10^6$ for $i \neq 1$; $b_1 = -1$, representing that the value of $b_1$ is unknown$)$.
The fourth line contains a single integer $q$ $(1 \le q \le 2 \cdot 10^{5})$ — the number of questions.
Each of the following $q$ lines contains a single integer $x_i$ $(1 \le x_i \le 10^6)$ — representing the $i$-th question.
Output $q$ integers — the answers to each of his $q$ questions.
Input
The first line contains a single integer $n$ $(1 \le n \le 2 \cdot 10^{5})$ — the size of arrays $a$ and $b$.
The second line contains $n$ integers $a_1, a_2, ..., a_n$ $(1 \le a_i \le 10^6)$.
The third line contains $n$ integers $b_1, b_2, ..., b_n$ $(1 \le b_i \le 10^6$ for $i \neq 1$; $b_1 = -1$, representing that the value of $b_1$ is unknown$)$.
The fourth line contains a single integer $q$ $(1 \le q \le 2 \cdot 10^{5})$ — the number of questions.
Each of the following $q$ lines contains a single integer $x_i$ $(1 \le x_i \le 10^6)$ — representing the $i$-th question.
Output
Output $q$ integers — the answers to each of his $q$ questions.
Samples
2
3 7
-1 5
3
1
4
3
2
4
2
6
2 5 4 1 3 6
-1 4 6 2 3 5
3
1
8
4
10
29
9
Note
Consider the first test case.
- $b_1 = 1$: We need to convert $[3, 7] \rightarrow [1, 5]$. We can perform the following operations:
$[3, 7]$ $\xrightarrow[\text{decrease}]{\text{i = 1}}$ $[2, 6]$ $\xrightarrow[\text{decrease}]{\text{i = 1}}$ $[1, 5]$
Hence the answer is $2$.
- $b_1 = 4$: We need to convert $[3, 7] \rightarrow [4, 5]$. We can perform the following operations:
$[3, 7]$ $\xrightarrow[\text{decrease}]{\text{i = 2}}$ $[3, 6]$ $\xrightarrow[\text{decrease}]{\text{i = 2}}$ $[3, 5]$ $\xrightarrow[\text{increase}]{\text{i = 1}}$ $[4, 6]$ $\xrightarrow[\text{decrease}]{\text{i = 2}}$ $[4, 5]$
Hence the answer is $4$.
- $b_1 = 3$: We need to convert $[3, 7] \rightarrow [3, 5]$. We can perform the following operations:
$[3, 7]$ $\xrightarrow[\text{decrease}]{\text{i = 2}}$ $[3, 6]$ $\xrightarrow[\text{decrease}]{\text{i = 2}}$ $[3, 5]$
Hence the answer is $2$.