FANDOM


In this post I will compare the following notations:

  • FGH
  • HH
  • SGH
  • BEAF
  • Bird's Arrays
  • Hyp Cos's strong Arrays
  • R function
  • EXE
  • HAN
  • My own bracket notation
  • AAN
  • AM12AN
  • My AN

Anything else you think I should add? Let me know.

Below n^n

Standard Form SGH HH FGH BEAF BAN SAN R ExE HAN Bracket AAN AMAN My AN
0 \(g_0(n)\) 0 0 0 0 0 0 0 0 0 0 0 0
a (a number) \(g_a(n)\) a a a a a a a a a a(a) a a
n \(g_\omega(n)\) \(H_0(n)\) n {n} {n} s(n) n n n n a(n) n n
n+a \(g_{\omega+a}(n)\) \(H_a(n)\) \({f_0}^n(n)\) {n}+a {n}+a s(n)+a n+a n+a n+a n+a a(n)+a n+a n+a
2n \(g_{\omega2}(n)\) \(H_\omega(n)\) \(f_1(n)\) 2*{n} 2*{n} 2*s(n) 2n 2n 2n 2n a(n,2) 2n n[0]2
3n \(g_{\omega3}(n)\) \(H_{\omega2}(n)\) 3n 3{n} 3{n} 3*s(n) 3n 3n 3n 3n a(n,3) 3n n[0]3
n^2 \(g_{\omega^2}(n)\) n^2 n^2 {n,2} {n,2} s(n,2) n^2 n^2 n^2 n^2 a(n,n) n(n,2) n[1]2 = n[0]n
2n^2 \(g_{\omega^22}(n)\) 2n^2 2n^2 2{n,2} 2{n,2} 2s(n,2) 2n^2 2n^2 2n^2 2n^2 a(n,2,1) 2*n(n,2) (n[1]2)[0]2
n^3 \(g_{\omega^3}(n)\) n^3 n^3 {n,3} {n,3} s(n,3) n^3 n^3 n^3 n^3 a(n,3,1) n(n,3) n[1]3
n^4 \(g_{\omega^4}(n)\) n^4 n^4 {n,4} {n,4} s(n,4) n^4 n^4 n^4 n^4 a(n,4,1) n(n,4) n[1]4

Up to n^^n

Standard SGH HH FGH BEAF BAN SAN R ExE HAN Bracket AAN AMAN My AN
n^n \(g_{\omega^\omega}(n)\) \(H_{\omega^2}(n)\) \(f_2(n)\) {n,n} {n,n} s(n,n) nR0 En n! n|() a(n,n,1) n(n,n) n[1]n
n^(n+1) \(g_{\omega^{\omega+1}}(n)\) \(H_{\omega^2+1}(n)\) \(f_2(f_0(n))\) {n,n+1} {n,n+1} s(n,n+1) (n+1)R0 E(n+1) (n+1)! (n+1)|() a(n,n,1) n(n,n+1) n[1](n+1)
n^(2n) \(g_{\omega^{\omega2}}(n)\) \(H_{\omega^2+\omega}(n)\) \(f_2(f_1(n))\) {n,2n} {n,2n} s(n,2n) (2n)R0 E(2n) (2n)! (2n)|() a(n,a(n,2),1) n(n,2n) n[1](2n)
n^n^2 \(g_{\omega^{\omega^2}}(n)\) \(H_{\omega^2+\omega}(n)\) \(f_2(f_1(n))\) {n,{n,2}} {n,{n,2}} s(n,s(n,2)) (n^2)R0 E(n^2) (n^2)! (n^2)|() a(n,2,2) n(n,n(n,2)) = n(n,2|2) n[1]n[1]2
n^^3 \(g_{\omega^{\omega^\omega}}(n)\) \(H_{\omega^22}(n)\) \(f_2(f_2(n))\) {n,3,2} {n,3,2} s(n,3,2) nR0R0 En#2 (n!)! n|()|() a(n,3,2) n(n,3|2) n[2]3
n^^4 \(g_{\omega^{\omega^{\omega^\omega}}}(n)\) \(H_{\omega^23}(n)\) \(f_2(f_2(f_2(n)))\) {n,4,2} {n,4,2} s(n,4,2) nR0R0R0 En#3 ((n!)!)! n|()|()|() a(n,4,2) n(n,4|2) n[2]4

Up to \(f_\omega(n)\)

Up-arrow Notation SGH HH FGH BEAF BAN SAN R ExE HAN
n^^n \(g_{\varepsilon_0}(n)\) \(H_{\omega^3}(n)\) \(f_3(n)\) {n,n,2} {n,n,2} s(n,n,2) nR1 En#n n!1
n^^(n+1) \(g_{\omega^{\varepsilon_0+1}}(n)\) \(H_{\omega^3+1}(n)\) \(f_3(f_0(n))\) {n,n+1,2} {n,n+1,2} s(n,n+1,2) (n+1)R1 #n#(n+1) (n+1)!1
n^^(2n) \(g_{\varepsilon_1}(n)\) \(H_{\omega^3+\omega}(n)\) \(f_3(f_1(n))\) {n,2n,2} {n,2n,2} s(n,2n,2) (2n)R1 En#(2n) (2n)!1
n^^n^2 \(g_{\varepsilon_\omega}(n)\) \(H_{\omega^3+\omega*2}(n)\) \(f_3(f_1(n))\) {n,{n,2},2} {n,{n,2},2} s(n,s(n,2),2) (n^2)R1 En#(n^2) (n^2)!1
n^^n^n \(g_{\varepsilon_{\omega^\omega}}(n)\) \(h_{\omega^3+\omega^2}(n)\) \(f_3(f_2(n))\) {n,{n,n},2} {n,{n,n},2} s(n,s(n,n),2) nR0R1 En#1#2 (n!)!1
n^^^3 (n)\) \(H_{{\omega^3}2}(n)\) \(f_3(f_3(n))\) {n,3,3} {n,3,3} s(n,3,3) nR1R1 En#n#2 (n!1)!1
n^^^n \(g_{\zeta_0}(n)\) \(H_{\omega^4}(n)\) \(f_4(n)\) {n,n,3} {n,n,3} s(n,n,3) nR2 En#n#n n!2
n^^^n^^n \(g_{\zeta_\varepsilon_0}(n)\) \(H_{\omega^4+\omega^3}(n)\) \(f_4(f_3(n))\) {n,{n,n,2},3} {n,{n,n,2},3} s(n,s(n,n,2),3) nR1R2 En#n#1#2 (n!1)!2
n^^^^3 \(g_{\zeta_\zeta_0}(n)\) \(H_{{\omega^4}2}(n)\) \(f_4(f_4(n))\) {n,3,4} {n,3,4} s(n,3,4) nR2R2 En#n#n#2 (n!2)!2
n^^^^n \(g_\varphi(3,0)(n)\) \(H_{\omega^5}(n)\) \(f_5(n)\) {n,n,4} {n,n,4} s(n,n,4) nR3 En#n#n#n n!3
n^^^^^n \(g_\varphi(4,0)(n)\) \(H_{\omega^6}(n)\) \(f_6(n)\) {n,n,5} {n,n,6} s(n,n,5) nR4 En#n#n#n#n n!4

Up to \(f_{\omega2}(n)\)

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