Ex - Dice Product 2

AtCoder

IDabc239_h

Time2000ms

Memory256MB

Difficulty—

Snuke has a die (singular of dice) that shows integers from $1$ through $N$ with equal probability, and an integer $1$. He repeats the following operation while his integer is less than or equal to $M$. * He rolls the die. If the die shows an integer $x$, he multiplies his integer by $x$. Find the expected value of the number of times he rolls the die until he stops, modulo $10^9+7$. Definition of the expected value modulo $10^9+7$ We can prove that the desired expected value is always a rational number. Moreover, under the constraints of the problem, when the value is represented as an irreducible fraction $\frac{P}{Q}$, we can also prove that $Q \not\equiv 0 \pmod{10^9+7}$. Thus, an integer $R$ such that $R \times Q \equiv P \pmod{10^9+7}$ and $0 \leq R \lt 10^9+7$ is uniquely determined. Answer such $R$. ## Constraints * $2 \leq N \leq 10^9$ * $1 \leq M \leq 10^9$ ## Input Input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

2 1

Output #1

2

The answer is the expected value of the number of rolls until it shows $2$ for the first time. Thus, $2$ should be printed.

Input #2

2 39

Output #2

12

The answer is the expected value of the number of rolls until it shows $2$ six times. Thus, $12$ should be printed.

Input #3

3 2

Output #3

250000004

The answer is $\frac{9}{4}$. We have $4 \times 250000004 \equiv 9 \pmod{10^9+7}$, so $250000004$ should be printed.  
Note that the answer should be printed modulo $\bf{10^9 + 7 = 1000000007}$.

Input #4

2392 39239

Output #4

984914531

Input #5

1000000000 1000000000

Output #5

776759630

API Response (JSON)

{
  "problem": {
    "name": "Ex - Dice Product 2",
    "description": {
      "content": "Snuke has a die (singular of dice) that shows integers from $1$ through $N$ with equal probability, and an integer $1$.   He repeats the following operation while his integer is less than or equal to ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc239_h"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Snuke has a die (singular of dice) that shows integers from $1$ through $N$ with equal probability, and an integer $1$.  \nHe repeats the following operation while his integer is less than or equal to ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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