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 Java (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.util.*;
import java.math.*;
import java.io.*;

class Main {
    private static Map<BigInteger, Integer> memo = new HashMap<>();

    private static int f(BigInteger n) {
	if (n.bitCount() == 0) return 0;

	// System.out.println("Bit count for n = " + n.bitCount() + " Number is " + n.toString());

	if (memo.containsKey(n)) {
	    // System.out.println("Memoed!");
	    return memo.get(n);
	} else {
	    memo.put(n, 1 + f(n.remainder(BigInteger.valueOf(n.bitCount()))));
	    return memo.get(n);
	}
    }

    public static void main(String[] args) throws Exception{
	BufferedReader reader = new BufferedReader(new InputStreamReader(System.in));
	int digits = Integer.parseInt(reader.readLine());
	BigInteger num = new BigInteger("0" + reader.readLine(), 2);
	for (int i = digits - 1; i >= 0; --i) {
	    // System.out.println("Trying with " + num.flipBit(i).toString());
	    System.out.println(f(num.flipBit(i)));
	}
    }
}