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I found some nice idea about how to calculate n(k) less directly. Suppose there is an ordinal hierarchy \(F_\alpha(n)\) (algorithm for evaluating it is unknown), and we have some strings which codes ordinals for this hierarchy:

String Ordinal
A 1
AA 2
An n
B w
BA w+1
BAA w+2
BAn w+n
BB w2
BBB w3
Bn wn
C w^2
CB w^2+w
CC (w^2)2
CCC (w^2)3
D w^3
E w^4

In general, plugging n-th letter gives ordinal w^(n-1).

We know that: for n(k) we can use only k letters, which limits at single (k+1)-th letter. So, \(n(k) = F_{\omega^\omega}(n)\) in the hierarchy described above. But the challenging part is inventing an algorithm for evaluate \(F_\alpha(n)\).

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