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 Ruby (2.7.1) (judge verdict: Accepted)

def prepare(n, mod, x)
  dp = Array.new(n, 1)
  return [0, 0, dp] if mod == 0
  score = 0
  (1 ... n).each do |i|
    dp[i] = (dp[i - 1] * 2) % mod
  end
  (1 .. n).each do |i|
    score += dp[n - i] if x[i - 1] == '1'
  end
  [mod, score, dp]
end

def f(x)
  for i in 0 .. 100
    return i if x == 0
    popcount = 32.times.count{|j| x[j] > 0 }
    x %= popcount
  end
end

N = gets.to_i
X = gets.chomp
popcount = X.count('1')
mod1, score1, dp1 = prepare(N, popcount + 1, X)
mod2, score2, dp2 = prepare(N, popcount - 1, X)

ans = (1 .. N).map do |i|
  if X[i - 1] == '0'
    f((score1 + dp1[N - i]) % mod1) + 1
  elsif popcount == 1
    0
  else
    f((score2 - dp2[N - i]) % mod2) + 1
  end
end
puts ans