Let $g(n)$ be a sequence defined as follows:
$g(4) = 13$,
$g(n) = g(n-1) + \gcd(n, g(n-1))$ for $n \gt 4$.
The first few values are:
| $n$ | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | ... |
| $g(n)$ | 13 | 14 | 16 | 17 | 18 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 51 | 54 | 55 | 60 | ... |
You are given that $g(1\,000) = 2524$ and $g(1\,000\,000) = 2624152$.
Find $g(8^N)$.