Squares

AtCoder

IDarc125_b

Time2000ms

Memory256MB

Difficulty—

You are given an integer $N$. Find the number of pairs of integers $(x, y)$ that satisfy the following conditions, modulo $998244353$. * $1 \leq x,y \leq N$. * $x^2-y$ is a square number. (Assume $0$ is also a square number.) ## Constraints * $1 \leq N \leq 10^{12}$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ [samples]

Samples

Input #1

Output #1

2

We have the following two pairs.

*   $x=1,y=1$
    
*   $x=2,y=3$

Input #2

Output #2

Input #3

10000000000

Output #3

52583544

API Response (JSON)

{
  "problem": {
    "name": "Squares",
    "description": {
      "content": "You are given an integer $N$. Find the number of pairs of integers $(x, y)$ that satisfy the following conditions, modulo $998244353$. *   $1 \\leq x,y \\leq N$.      *   $x^2-y$ is a square number. (A",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc125_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given an integer $N$. Find the number of pairs of integers $(x, y)$ that satisfy the following conditions, modulo $998244353$.\n\n*   $1 \\leq x,y \\leq N$.\n    \n*   $x^2-y$ is a square number. (A...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments