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 C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;

namespace _20200711
{
    class Program
    {
        static void Main()
        {
            var n = int.Parse(Console.ReadLine());
            var x_ = Console.ReadLine();
            var x = new int[n];
            var popx = 0;
            for (var i = 0; i < n; ++i)
            {
                x[i] = int.Parse(x_.Substring(i, 1));
                if (x[i] == 1)
                {
                    ++popx;
                }
            }

            var n_pn_plus = new int[n];
            var n_pn_minus = new int[n];
            var minusValid = popx > 1;
            var t_plus = 1;
            var t_minus = 1;
            for (var i = 0; i < n; ++i)
            {
                t_plus %= popx + 1;
                n_pn_plus[i] = t_plus;
                t_plus *= 2;
                if (minusValid)
                {
                    t_minus %= popx - 1;
                    if (t_minus < 0)
                    {
                        t_minus += popx - 1;
                    }
                    n_pn_minus[i] = t_minus;
                    t_minus *= 2;
                }
            }

            var npnFirstPlus = 0;
            var npnFirstMinus = 0;
            for (var i = 0; i < n; ++i)
            {
                npnFirstPlus = (npnFirstPlus * 2 + x[i]) % (popx + 1);
                if (minusValid)
                {
                    npnFirstMinus = (npnFirstMinus * 2 + x[i]) % (popx - 1);
                }
            }

            for (var i = 0; i < n; ++i)
            {
                var pos = n - i - 1;
                if (x[i] == 1)
                {
                    if (popx == 1)
                    {
                        Console.WriteLine(0);
                        continue;
                    }
                    var npn = (npnFirstMinus - n_pn_minus[pos]) % (popx - 1);
                    if (npn < 0)
                    {
                        npn += (popx - 1);
                    }
                    Console.WriteLine(F(npn) + 1);
                }
                else
                {
                    Console.WriteLine(F((npnFirstPlus + n_pn_plus[pos]) % (popx + 1)) + 1);
                }
            }
        }

        static int F(int x)
        {
            var tmp = x;
            var count = 0;
            while (tmp > 0)
            {
                tmp %= P(tmp);
                ++count;
            }
            return count;
        }

        static int P(int x)
        {
            var tmp = x;
            var count = 0;
            while (tmp > 0)
            {
                if (tmp % 2 == 1)
                {
                    ++count;
                }
                tmp /= 2;
            }
            return count;
        }
    }
}