Weak Goodstein sequence (noch nicht übersetzt)

For any positive integer n, the nth weak Goodstein sequence {g₁, g₂, g₃, ...} is defined as:

• g₁ = 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 6th weak Goodstein sequence is {6, 11, 17, 25, ...}:

• g₁ = 6.
• g₂ = 11 since 6 = 110₂, 110₃ = 12, and 12 - 1 = 11.
• g₃ = 17 since 11 = 102₃, 102₄ = 18, and 18 - 1 = 17.
• g₄ = 25 since 17 = 101₄, 101₅ = 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 nth weak Goodstein sequence.
It can be verified that G(2) = 3, G(4) = 21 and G(6) = 381.
It can also be verified that ∑ G(n) = 2517 for 1 ≤ n < 8.

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