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I've decided to explore Friedman's stuff better. I should start with n(k) because it's slowest widely known function defined by him.

Let's define the function L(n,m) as the number of Friedman strings which have n letters from alphabet {1,2,...,m}.

For example, let's compute L(4,2). First, we write all strings with n letters from alphabet {1,2,...,m}:

1111

1112

1121

1122

1211

1212

1221

1222

2111

2112

2121

2122

2211

2212

2221

2222

Then we remove strings written in bold text from this list because they're not Friedman. It contains 8 strings and so L(4,2) = 8.

Now we can redefine n(k) as the largest p so that L(p,k) > 0. The problem drops to explore the behavior of L(p,k) function.

For k = 2, we have the following sequence:

L(1,2) = 2

L(2,2) = 4

L(3,2) = 8

L(4,2) = 8

L(5,2) = 16

L(6,2) = 12

L(7,2) = 24

L(8,2) = 4

L(9,2) = 8

L(10,2) = 2

L(11,2) = 4

L(12,2) = 0

L(i,2) = 0 (for i > 12)

We can note that if m is even, then L(m+1,n) = L(m,n)*n. If m is odd, then it's not clear how L(m+1,n) compares to L(m,n). If we find that out, bounding n(k) might be straightforward.

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