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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)

See alsoEdit

Googology in Asia

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Pair sequence number · Bashicu matrix system
Indian counting system: Lakh · Crore · Uppala · Bodhisattva
Chinese and Japanese counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Buddhist text: Tallakshana · Dvajagravati · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra
Other: Taro's multivariable Ackermann function · Sushi Kokuuhen

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