# Problem 396: Weak Goodstein sequence
For any positive integer n, the nth weak Goodstein sequence {g1, g2, g3,
...} is defined as: g1 = n for k > 1, gk is obtained by writing gk-1
in base k, interpreting it as a base k + 1 number, and subtracting 1.
The sequence terminates when gk 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 ΣG(n) = 2517 for 1 ≤ n < 8. Find
the last 9 digits of ΣG(n) for 1 ≤ n < 16.