Just Arrange the Icons
BerPhone X is almost ready for release with $$$n$$$ applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between $$$1$$$ and $$$n$$$, inclusive; the $$$i$$$-th application has category $$$c_i$$$. You can choose $$$m$$$ — the number of screens and $$$s$$$ — the size of each screen. You need to fit all $$$n$$$ icons of the applications (one icon representing one application) meeting the following requirements:  On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);  Each screen must be either completely filled with icons (the number of icons on the screen is equal to $$$s$$$) or almost filled with icons (the number of icons is equal to $$$s-1$$$). Your task is to find the minimal possible number of screens $$$m$$$.
In the first test case of the example, all the icons can be placed on three screens of size $$$4$$$: a screen with $$$4$$$ icons of the category $$$1$$$, a screen with $$$3$$$ icons of the category $$$1$$$, and a screen with $$$4$$$ icons of the category $$$5$$$.
The first line contains an integer $$$t$$$ ($$$1 \le t \le 10\,000$$$) — the number of test cases in the input. Then $$$t$$$ test cases follow.The first line of each test case contains an integer $$$n$$$ ($$$1 \le n \le 2\cdot10^6$$$) — the number of the icons. The second line contains $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$1 \le c_i \le n$$$), where $$$c_i$$$ is the category of the $$$i$$$-th application.It is guaranteed that the sum of the values of $$$n$$$ for all test cases in the input does not exceed $$$2\cdot10^6$$$.
Print $$$t$$$ integers — the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer $$$m$$$ — the minimum number of screens on which all $$$n$$$ icons can be placed satisfying the given requirements.
import sys
input = sys.stdin.readline
res = []
t = int(input())
for _ in range(t):
    n = int(input())
    l = map(int, input().split())
    c = [0] * n
    for v in l:
        c[v - 1] += 1
    c.sort()
    c.reverse()
    while c[-1] == 0:
        c.pop()
    best = n
    for i in range(2, c[-1] + 2):
        out = 0
        for v in c:
            smol = -v % i
            tol = (v + smol) // i - smol
            if tol >= 0:
                out += smol + tol
            else:
                out += n
        best = min(best, out)
    res.append(best)
print('\n'.join(map(str, res)))
import sys
input = sys.stdin.readline
res = []
t = int(input())
for _ in range(t):
    n = int(input())
    l = map(int, input().split())
    c = [0] * n
    for v in l:
        c[v - 1] += 1
    c.sort()
    c.reverse()
    while c[-1] == 0:
        c.pop()
    best = n
    for i in range(2, c[-1] + 2):
        out = 0
        for v in c:
            smol = -v % i
            tol = (v + smol) // i - smol
            if tol >= 0:
                out += smol + tol
            else:
                out += n
        best = min(best, out)
    res.append(best)
print('\n'.join(map(str, res)))
t = int(input())
for i in range(t):
    n = int(input())
    c = [int(x) for x in input().split()]
    dt = [0] * n
    for j in c:
        dt[j - 1] += 1
    dt = sorted(dt)
    for i in range(len(dt)):
        if dt[i] != 0:
            dt = dt[i:]
            break
    for k in range(1, dt[0] + 1):
        apps = dt[0] // k
        tr = True
        sums = 0
        for i in dt:
            mn = i // (apps + 1)
            sums += mn
            r = i % (apps + 1)
            if r > 0 and r + mn >= apps:
                sums += 1
            elif r > 0:
                tr = False
                break
        if tr:
            print(sums)
            break
        tr = True
        sums = 0
        for i in dt:
            mn = i // apps
            sums += mn
            r = i % apps
            if r > 0 and r + mn >= apps - 1:
                sums += 1
            elif r > 0:
                tr = False
                break
        if tr:
            print(sums)
            break