Weak Goodstein sequence

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.

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.

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 ΣG(n) = 2517 for 1 ≤ n < 8.

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

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter