## FANDOM

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The N-growing hierarchy is a hierarchy/notation based on the fast-growing hierarchy, created by Japanese googologist Aeton (2013) [1].

### Definition Edit

• $$[m]_0(n) = m\times n$$
• $$[m]_{\alpha+1}(n) = [m]^{n-1}_\alpha(m)$$, and if $$n=1$$, $$[m]_{\alpha+1}(1)=[m]^0_\alpha(m)=m$$
• $$[m]_\alpha(n) = [m]_{\alpha[n]}(m)$$, when $$\alpha$$ is a limit ordinal and $$\alpha[n]$$ is the $$n$$th term of fundamental sequence assigned to ordinal $$\alpha$$.

And when $$m=10$$, it can be called 10-growing hierarchy. And similarly, 3-growing hierarchy, 16-growing hierarchy, or Googol-growing hierarchy are also possible.

However, If $$m=n$$, it is called Diagonal n-growing hierarchy and its notation changes as follows.

• $$(N_\alpha(n) = [n]_\alpha(n))$$
• $$N_0(n) = n\times n=n^2$$
• $$N_{\alpha+1}(n) = N^{n-1}_\alpha(n)$$
• $$N_\alpha(n) = N_{\alpha[n]}(n)$$

### Examples Edit

This function is exactly equal to up-arrow notation, and probably array notation, but for that reason, when $$m=2$$ and $$\alpha\geq\omega$$, it does not grow well.

• $$[16]_4(8) = 16\uparrow^4 8$$
• $$[10]_{\omega+1}(100) = \{10,100,1,2\}=$$ Corporal
• $$[3]^{64}_{\omega}(4)$$ = Graham's number $$\lesssim[4]_{\omega+1}(65) = \{4,65,1,2\}$$
• $$[4]_{\omega^2+1}(64) = \{4,64,1,1,2\}<$$ Fish number 1
• $$N_\omega(3) = [3]_3(3) = 3\uparrow^3 3=$$ Tritri
• $$N_{\omega^2}(10) = \{10,10,10,10\}=$$ General

Because of the reason that $$[m]_{\omega^\omega}(n)=\{m,n+2(1)2\}$$, this function doesn't match exactly over $$\{m,n(1)2\}$$ level of BEAF, in $$\alpha\geq\omega^\omega$$ level.

• $$N_{\omega^{98}}(10) = [10]_{\omega^\omega}(98) = \{10,100 (1) 2\}=$$ Goobol
• $$[10]_{\omega^\omega}([10]_{\omega^\omega}(98)-2)=$$ goobolplex $$\approx[10]^2_{\omega^\omega}(98)$$

### Sources Edit

1. n-growing hierarchy (Japanese Page)