Berland Regional
Contest: Educational Codeforces Round 108 (Rated for Div. 2)
Difficulty rating: 1400
Tags: brute force, data structures, greedy, number theory, sortings

Polycarp is an organizer of a Berland ICPC regional event. There are $$$n$$$ universities in Berland numbered from $$$1$$$ to $$$n$$$. Polycarp knows all competitive programmers in the region. There are $$$n$$$ students: the $$$i$$$-th student is enrolled at a university $$$u_i$$$ and has a programming skill $$$s_i$$$.

Polycarp has to decide on the rules now. In particular, the number of members in the team.

Polycarp knows that if he chooses the size of the team to be some integer $$$k$$$, each university will send their $$$k$$$ strongest (with the highest programming skill $$$s$$$) students in the first team, the next $$$k$$$ strongest students in the second team and so on. If there are fewer than $$$k$$$ students left, then the team can't be formed. Note that there might be universities that send zero teams.

The strength of the region is the total skill of the members of all present teams. If there are no teams present, then the strength is $$$0$$$.

Help Polycarp to find the strength of the region for each choice of $$$k$$$ from $$$1$$$ to $$$n$$$.

Input format:
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of testcases.

The first line of each testcase contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of universities and the number of students.

The second line of each testcase contains $$$n$$$ integers $$$u_1, u_2, \dots, u_n$$$ ($$$1 \le u_i \le n$$$) — the university the $$$i$$$-th student is enrolled at.

The third line of each testcase contains $$$n$$$ integers $$$s_1, s_2, \dots, s_n$$$ ($$$1 \le s_i \le 10^9$$$) — the programming skill of the $$$i$$$-th student.

The sum of $$$n$$$ over all testcases doesn't exceed $$$2 \cdot 10^5$$$.

Output format:
For each testcase print $$$n$$$ integers: the strength of the region — the total skill of the members of the present teams — for each choice of team size $$$k$$$.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
In the first testcase the teams from each university for each $$$k$$$ are:

- $$$k=1$$$:   university $$$1$$$: $$$[6], [5], [5], [3]$$$;  university $$$2$$$: $$$[8], [1], [1]$$$;
- $$$k=2$$$:   university $$$1$$$: $$$[6, 5], [5, 3]$$$;  university $$$2$$$: $$$[8, 1]$$$;
- $$$k=3$$$:   university $$$1$$$: $$$[6, 5, 5]$$$;  university $$$2$$$: $$$[8, 1, 1]$$$;
- $$$k=4$$$:   university $$$1$$$: $$$[6, 5, 5, 3]$$$;

Example input:
4
7
1 2 1 2 1 2 1
6 8 3 1 5 1 5
10
1 1 1 2 2 2 2 3 3 3
3435 3014 2241 2233 2893 2102 2286 2175 1961 2567
6
3 3 3 3 3 3
5 9 6 7 9 7
1
1
3083

Example output:
29 28 26 19 0 0 0 
24907 20705 22805 9514 0 0 0 0 0 0 
43 43 43 32 38 43 
3083

Editorial:
There are two important observations to make.
The first one is that you can calculate the answers for each university independently of each other and sum them up to obtain the true answer.
The second one is that if there are $$$x$$$ students in an university, then that university can only contribute to answers for $$$k$$$ from $$$1$$$ to $$$x$$$.
So if we learn to calculate the contribution of the $$$i$$$-th university for some fixed $$$k$$$ in $$$O(1)$$$, then we will be able to iterate over all possible $$$k$$$ for each university and get the solution in $$$O(\sum \limits_{i=1}^{n} x_i) = O(n)$$$, where $$$x_i$$$ is the number of students in the $$$i$$$-th university.
To achieve it, you have to gather the sum of the maximum number of students that can form full teams of size $$$k$$$. That must be the highest number less than or equal to $$$x_i$$$ that is divisible by $$$k$$$, so $$$\lfloor \frac{x_i}{k} \rfloor \cdot k$$$. Sort the students of each university, precalculate partial sums, and now you are free to add the prefix sum of that number of students to the answer for $$$k$$$.
Overall complexity: $$$O(n \log n)$$$ per testcase.