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)

class Integer
  def popcount
    to_s(2).count('1')
  end
end

def f(n)
  c = 0
  while n > 0
    n %= n.popcount
    c += 1
  end
  c
end

def modpow(base, n, mod)
  result = 1
  while n > 0
    if n.odd?
      result = result * base % mod
    end

    base = (base * base) % mod

    n /= 2
  end
  result % mod
end

n = gets.to_i
x = gets.chomp
x_int = x.to_i(2)
base_popcount = x.count('1')

mod_zero = x_int % (base_popcount + 1)
mod_one  = x_int % (base_popcount - 1) unless base_popcount == 1

0.upto(n - 1) do |index|
  if x[index] == '1' && base_popcount == 1
    puts 0
    next
  end

  shift = n - index - 1
  mod =
    if x[index] == '0'
      popcount = base_popcount + 1
      (mod_zero + modpow(2, shift, popcount)) % popcount
    else
      popcount = base_popcount - 1
      (mod_one  - modpow(2, shift, popcount)) % popcount
    end

  puts f(mod) + 1
end