Even XOR

AtCoder

IDarc146_c

Time2000ms

Memory256MB

Difficulty—

How many sets $S$ consisting of non-negative integers between $0$ and $2^N-1$ (inclusive) satisfy the following condition? Print the count modulo $998244353$. * Every non-empty subset $T$ of $S$ satisfies at least one of the following: * The number of elements in $T$ is odd. * The $\mathrm{XOR}$ of the elements in $T$ is not zero. What is $\mathrm{XOR}$?The bitwise $\mathrm{XOR}$ of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows: * When $A \oplus B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of the digits in that place of $A$ and $B$ is $1$, and $0$ otherwise. For example, we have $3 \oplus 5 = 6$ (in base two: $011 \oplus 101 = 110$). Generally, the bitwise $\mathrm{XOR}$ of $k$ non-negative integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1 \oplus p_2) \oplus p_3) \oplus \dots \oplus p_k)$. We can prove that this value does not depend on the order of $p_1, p_2, p_3, \dots, p_k$. ## Constraints * $1 \le N \le 2 \times 10^5$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ [samples]

Samples

Input #1

Output #1

15

Sets such as $\lbrace 0,2,3 \rbrace$, $\lbrace 1 \rbrace$, and $\lbrace \rbrace$ satisfy the condition.
On the other hand, $\lbrace 0,1,2,3 \rbrace$ does not.
This is because the subset $\lbrace 0,1,2,3 \rbrace$ of $\lbrace 0,1,2,3 \rbrace$ has an even number of elements, whose bitwise $\mathrm{XOR}$ is $0$.

Input #2

Output #2

743874490

API Response (JSON)

{
  "problem": {
    "name": "Even XOR",
    "description": {
      "content": "How many sets $S$ consisting of non-negative integers between $0$ and $2^N-1$ (inclusive) satisfy the following condition? Print the count modulo $998244353$. *   Every non-empty subset $T$ of $S$ sa",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc146_c"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "How many sets $S$ consisting of non-negative integers between $0$ and $2^N-1$ (inclusive) satisfy the following condition? Print the count modulo $998244353$.\n\n*   Every non-empty subset $T$ of $S$ sa...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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