---
id: 5900f4f81000cf542c51000b
title: 'Problem 396: Weak Goodstein sequence'
challengeType: 5
forumTopicId: 302061
dashedName: problem-396-weak-goodstein-sequence
---

# --description--

For any positive integer $n$, the $n$th weak Goodstein sequence $\\{g1, g2, g3, \ldots\\}$ is defined as:

- $g_1 = n$
- for $k > 1$, $g_k$ is obtained by writing $g_{k - 1}$ in base $k$, interpreting it as a base $k + 1$ number, and subtracting 1.

The sequence terminates when $g_k$ becomes 0.

For example, the $6$th weak Goodstein sequence is $\\{6, 11, 17, 25, \ldots\\}$:

- $g_1 = 6$.
- $g_2 = 11$ since $6 = 110_2$, $110_3 = 12$, and $12 - 1 = 11$.
- $g_3 = 17$ since $11 = 102_3$, $102_4 = 18$, and $18 - 1 = 17$.
- $g_4 = 25$ since $17 = 101_4$, $101_5 = 26$, and $26 - 1 = 25$.

and so on.

It can be shown that every weak Goodstein sequence terminates.

Let $G(n)$ be the number of nonzero elements in the $n$th weak Goodstein sequence.

It can be verified that $G(2) = 3$, $G(4) = 21$ and $G(6) = 381$.

It can also be verified that $\sum G(n) = 2517$ for $1 ≤ n < 8$.

Find the last 9 digits of $\sum G(n)$ for $1 ≤ n < 16$.

# --hints--

`weakGoodsteinSequence()` should return `173214653`.

```js
assert.strictEqual(weakGoodsteinSequence(), 173214653);
```

# --seed--

## --seed-contents--

```js
function weakGoodsteinSequence() {

  return true;
}

weakGoodsteinSequence();
```

# --solutions--

```js
// solution required
```