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N function is the strongest function ever I made. It's inspired by R function.

For a positive integer n and non-negative integers m and h, we define a positive integer \(n\text{N}^{h}m\) in the following recursive way:

1. If \(h = 0\), then \(n\text{N}^{h}m = m\).

2. Suppose \(h = 1\).

 2-1. if \(m=0\), then \(n\text{N}^{h}m = n+1\).

 2-2. if \(m>0\), then \(n\text{N}^{h}m = n\text{N}^{n}(m-1)\).

3. If h > 1, then \(n\text{N}^{h}m = n\text{N}^{1}(n\text{N}^{h-1}m)\).

4. For a positive integer n, we define a positive integer \(n\text{N}\{0\}\) as \(n\text{N}^{1}n\).

So in the basic level, the growth rate is \(f_{\omega}(n)\)



5. nN{m+1}=nN{m}{m}...{m}(n times)

You can apply rule 4 and 5 inside of brace like {{0}}={0{0}}, {{0}}{{0}}...{{0}}={1{0}}, {n{0}}={{0}{0}}

6. nN{{...{0}...}}=nN{0,1}

unlike R function, nN{{...{0}...}}{0,1}=nN{0,1}{0,1}. So it is way weaker than R function

7. nN{m,1}{m,1}...{m,1}=nN{m+1,1}.

And we can use rule 3,4,5,6 again. Then, the limit is nN{{...{{0,1},1}...},1}.

7. nN{{...{{0,1},1}...},1}=nN{{{0,1}},1}=nN{{0{0,1}},1}

8.nN{{m+1{0,1},1}=nN{{m{0,1}{m{0,1}...{m{0,1},1}

Use rule 3,4,5,6.7.8 again.

9. nN{{{{...{{{{0,1}},1}}...}},1}}=nN{{{{0,1}}},1}

10. nN{{{...{0,1)...}},1}=nN{0,2}

We can also make nN{0,{0}}, nN{0,{0,1}}, nN{0,{{0,1}}}, nN{0.{{...{0,1}...}}}=nN{0,0,1}, nN{0,0,0,1}, and so on.

Also, I have more Idea of N function beyond nN{0,0,...,1}.

nN{0,0,0,...,1}=nN{0{0*}0}

nN{0{0*}0}{0{0*}0}...{0{0*}0}=nN{1{0*}0}

nN{n{0*}0}=nN{{0}{0*}0}

nN{0,1{0*}0}=nN{{{...{0{0*}0}...{0*}0}{0*}0}{0*}0}

nN{0,0,0,,1{0*}0}=nN{0{0*}1}

nN{0{0*}0,0,...,0,1}=nN{0{0*}0{0*}0}

nN{0{0*}0{0*}0...0{0*}0}=nN{0{0*}{0*}0}

nN{0{0*}{0*}...{0*}0}=nN{0{1*}0}

nN{0{n*}0}=nN{0{{0}*}0}

The limit is nN{0{0{..{0{0*}0}...*}0*}0}=nN{0*,1}.

And we can also consider like nN{0*,0*,1}, nN{0{0**}0}, nN{0**,1},.. etc.


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