Sequence A268687: a(n) = MAX(g_k(n)) weak Goodstein

A Goodstein sequence, using the "weak" or slower-growing definition, is described in OEIS sequence A266202:

A nonnegative n in ordinary (depth-1) base-k representation, is n rewritten as linear combination of powers of k:

      n = n₁×b^m₁ +...+ n_k×b^m_k
         where 0 < n_i < b
         and m₁ > ... > m_k >= 0.

For instance the ordinary representation of 34 in base 3 is 3³+2×3+1.

Let b_k(n) be the function that substitutes the bases of the base-k representation of n with the base k+1. For example, b₃(34) = b₃(3³+2×3+1) = 4³+2×4+1 = 73.

Define the weak Goodstein function as:

g_k(n) = n for k=0
b_k+1(g_k-1(n))-1 for k>0

See example (in A266202) for instances.

Let n be a fixed nonnegative: Goodstein's theorem shows that the sequence g_k(n) eventually stabilises and then decreases by 1 in each step until it reaches 0. Thereafter, all the values of g_k(n) less than 0 are not part of the sequence.

By Goodstein's theorem we conclude that g_k(n) is a finite sequence.

Sequence A268687 gives the maximum value seen in the Goodstein sequence starting with n.

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