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Weak Goodstein sequence

Problem 396

Published on 30 September 2012 at 02:00 am [Server Time]

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.


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