--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.

assert.strictEqual(weakGoodsteinSequence(), 173214653);

--seed--

--seed-contents--

function weakGoodsteinSequence() {

  return true;
}

weakGoodsteinSequence();

--solutions--

// solution required