Coincidence

AtCoder

IDabc138_f

Time2000ms

Memory256MB

Difficulty—

Given are integers $L$ and $R$. Find the number, modulo $10^9 + 7$, of pairs of integers $(x, y)$ $(L \leq x \leq y \leq R)$ such that the remainder when $y$ is divided by $x$ is equal to $y \text{ XOR } x$. What is $\text{ XOR }$?The XOR of integers $A$ and $B$, $A \text{ XOR } B$, is defined as follows: * When $A \text{ XOR } B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if either $A$ or $B$, but not both, has $1$ in the $2^k$'s place, and $0$ otherwise. For example, $3 \text{ XOR } 5 = 6$. (In base two: $011 \text{ XOR } 101 = 110$.) ## Constraints * $1 \leq L \leq R \leq 10^{18}$ ## Input Input is given from Standard Input in the following format: $L$ $R$ [samples]

Samples

Input #1

2 3

Output #1

3

Three pairs satisfy the condition: $(2, 2)$, $(2, 3)$, and $(3, 3)$.

Input #2

10 100

Output #2

Input #3

1 1000000000000000000

Output #3

68038601

Be sure to compute the number modulo $10^9 + 7$.

API Response (JSON)

{
  "problem": {
    "name": "Coincidence",
    "description": {
      "content": "Given are integers $L$ and $R$. Find the number, modulo $10^9 + 7$, of pairs of integers $(x, y)$ $(L \\leq x \\leq y \\leq R)$ such that the remainder when $y$ is divided by $x$ is equal to $y \\text{ XO",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc138_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given are integers $L$ and $R$. Find the number, modulo $10^9 + 7$, of pairs of integers $(x, y)$ $(L \\leq x \\leq y \\leq R)$ such that the remainder when $y$ is divided by $x$ is equal to $y \\text{ XO...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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