Anything Goes to Zero

AtCoder

IDaising2020_d

Time2000ms

Memory256MB

Difficulty—

Let $\mathrm{popcount}(n)$ be the number of `1`s 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: $N$ $X$ [samples]

Samples

Input #1

3
011

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, $f(2) = 1$.

Input #2

23
00110111001011011001110

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Anything Goes to Zero",
    "description": {
      "content": "Let $\\mathrm{popcount}(n)$ be the number of `1`s 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",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "aising2020_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Let $\\mathrm{popcount}(n)$ be the number of `1`s in the binary representation of $n$. For example, $\\mathrm{popcount}(3) = 2$, $\\mathrm{popcount}(7) = 3$, and $\\mathrm{popcount}(0) = 0$.\nLet $f(n)$ be...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments