Problem:
Score : 400 points

Problem Statement

Let \mathrm{popcount}(n) be the number of 1s in the binary representation of n.
For example, \mathrm{popcount}(3) = 2, \mathrm{popcount}(7) = 3, and \mathrm{popcount}(0) = 0.

Let f(n) be the number of times the following operation will be done when we repeat it until n becomes 0: "replace n with the remainder when n is divided by \mathrm{popcount}(n)." (It can be proved that, under the constraints of this problem, n always becomes 0 after a finite number of operations.)

For example, when n=7, it becomes 0 after two operations, as follows:

\mathrm{popcount}(7)=3, so we divide 7 by 3 and replace it with the remainder, 1.

\mathrm{popcount}(1)=1, so we divide 1 by 1 and replace it with the remainder, 0.

You are given an integer X with N digits in binary.
For each integer i such that 1 \leq i \leq N, let X_i be what X becomes when the i-th bit from the top is inverted.
Find f(X_1), f(X_2), \ldots, f(X_N).

Constraints

1 \leq N \leq 2 \times 10^5

X is an integer with N digits in binary, possibly with leading zeros.

Input

Input is given from Standard Input in the following format:

N
X

Output

Print N lines. The i-th line should contain the value f(X_i).

Sample Input 1

3
011

Sample Output 1

2
1
1

X_1 = 7, which will change as follows: 7 \rightarrow 1 \rightarrow 0. Thus, f(7) = 2.

X_2 = 1, which will change as follows: 1 \rightarrow 0. Thus, f(1) = 1.

X_3 = 2, which will change as follows: 2 \rightarrow 0. Thus, f(2) = 1.

Sample Input 2

23
00110111001011011001110

Sample Output 2

2
1
2
2
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
3

Solution in Python (3.8.2) (judge verdict: Runtime Error)

import numpy as np
N = int(input())
B = str(input())

B_array = np.array(list(map(int, list(B))))
B_1 = B_array.sum()
mod_list_m1 = []
mod_list = []
mod_list_p1 = []
for i in range(N):
    mod_list.append((2**i) % (B_1))
    mod_list_m1.append((2**i) % (B_1-1))
    mod_list_p1.append((2**i) % (B_1+1))
mod_list = np.array(mod_list)[::-1]
mod_list_m1 = np.array(mod_list_m1)[::-1]
mod_list_p1 = np.array(mod_list_p1)[::-1]

mod_m1 = B_array.dot(mod_list_m1) % (B_1-1)
mod_p1 = B_array.dot(mod_list_p1) % (B_1+1)

def calc_small(n, iter):
    iter += 1
    mod = n % bin(n).count("1")
    if mod == 0:
        return iter
    else:
        return calc_small(mod, iter)

for i in range(N):
    mod = (mod_m1 - mod_list_m1[i]) % (B_1-1) if B_array[i] == 1 else (mod_p1 + mod_list_p1[i]) % (B_1+1)
    iter = 1
    if mod == 0:
        print(iter)
    else:
        print(calc_small(mod, iter))