---
id: 5900f5141000cf542c510027
title: 'Problem 423: Consecutive die throws'
challengeType: 5
forumTopicId: 302093
dashedName: 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: (1,1,5,6,6,6,3) (1,1,5,6,6,6,3) (1,1,5,6,6,6,3) 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).1 For example, C(3) = 216, C(4) = 1290, C(11) = 361912500 and C(24) = 4727547363281250000.

Define S(L) as ∑ C(n) for 1 ≤ n ≤ L. For example, S(50) mod 1 000 000 007 = 832833871.

Find S(50 000 000) mod 1 000 000 007.

1 π denotes the prime-counting function, i.e. π(n) is the number of primes ≤ n.

# --hints--

`euler423()` should return 653972374.

```js
assert.strictEqual(euler423(), 653972374);
```

# --seed--

## --seed-contents--

```js
function euler423() {

  return true;
}

euler423();
```

# --solutions--

```js
// solution required
```