Division by Two with Something

AtCoder

IDagc039_c

Time2000ms

Memory256MB

Difficulty—

Given are integers $N$ and $X$. For each integer $k$ between $0$ and $X$ (inclusive), find the answer to the following question, then compute the sum of all those answers, modulo $998244353$. * Let us repeat the following operation on the integer $k$. Operation: if the integer is currently odd, subtract $1$ from it and divide it by $2$; otherwise, divide it by $2$ and add $2^{N-1}$ to it. How many operations need to be performed until $k$ returns to its original value? (The answer is considered to be $0$ if $k$ never returns to its original value.) ## Constraints * $1 \leq N \leq 2\times 10^5$ * $0 \leq X < 2^N$ * $X$ is given in binary and has exactly $N$ digits. (In case $X$ has less than $N$ digits, it is given with leading zeroes.) * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $X$ [samples]

Samples

Input #1

3
111

Output #1

40

For example, for $k=3$, the operation changes $k$ as follows: $1,0,4,6,7,3$. Therefore the answer for $k=3$ is $6$.

Input #2

6
110101

Output #2

Input #3

30
001110011011011101010111011100

Output #3

549320998

API Response (JSON)

{
  "problem": {
    "name": "Division by Two with Something",
    "description": {
      "content": "Given are integers $N$ and $X$. For each integer $k$ between $0$ and $X$ (inclusive), find the answer to the following question, then compute the sum of all those answers, modulo $998244353$. *   Let",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc039_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given are integers $N$ and $X$. For each integer $k$ between $0$ and $X$ (inclusive), find the answer to the following question, then compute the sum of all those answers, modulo $998244353$.\n\n*   Let...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments