1.5 KiB
1.5 KiB
| id | title | challengeType | forumTopicId | dashedName |
|---|---|---|---|---|
| 5900f5141000cf542c510027 | Problem 423: Consecutive die throws | 5 | 302093 | problem-423-consecutive-die-throws |
--description--
Let n be a positive integer.
A 6-sided die is thrown n times. Let c be the number of pairs of consecutive throws that give the same value.
For example, if n = 7 and the values of the die throws are (1, 1, 5, 6, 6, 6, 3), then the following pairs of consecutive throws give the same value:
\begin{align}
& (\underline{1}, \underline{1}, 5, 6, 6, 6, 3) \\\\
& (1, 1, 5, \underline{6}, \underline{6}, 6, 3) \\\\
& (1, 1, 5, 6, \underline{6}, \underline{6}, 3)
\end{align}$$
Therefore, $c = 3$ for (1, 1, 5, 6, 6, 6, 3).
Define $C(n)$ as the number of outcomes of throwing a 6-sided die $n$ times such that $c$ does not exceed $π(n)$.<sup>1</sup>
For example, $C(3) = 216$, $C(4) = 1290$, $C(11) = 361\\,912\\,500$ and $C(24) = 4\\,727\\,547\\,363\\,281\\,250\\,000$.
Define $S(L)$ as $\sum C(n)$ for $1 ≤ n ≤ L$.
For example, $S(50)\bmod 1\\,000\\,000\\,007 = 832\\,833\\,871$.
Find $S(50\\,000\\,000)\bmod 1\\,000\\,000\\,007$.
<sup>1</sup> $π$ denotes the prime-counting function, i.e. $π(n)$ is the number of primes $≤ n$.
# --hints--
`consecutiveDieThrows()` should return `653972374`.
```js
assert.strictEqual(consecutiveDieThrows(), 653972374);
```
# --seed--
## --seed-contents--
```js
function consecutiveDieThrows() {
return true;
}
consecutiveDieThrows();
```
# --solutions--
```js
// solution required
```