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Hexteto

Visualization of boogasix using exponentiation

Boogasix is equal to \(6\uparrow\uparrow\uparrow\uparrow6\), or six hexated to 6.

Googology Wiki user Hyp cos call this number hexteto, and it's equal to s(6,6,4) = s(6,2,5) = s(6,4,1,2) in strong array notation.[1]

Boboris02 calls this number Hexar, which also equals \(Q_{1,0}(6)=Q_{6}(6)\) in the Q-supersystem.[2]

Etymology

The name "boogasix" is formed by applying the booga- prefix to 6.

The name "hexteto" is based on the word "hexa" and "tetra" (Greek for 6 and 4), meaning a base-6 number, generated using hyper-4 operation based on exponentiation.

The name "Hexar" comes from the greek word "hexa" meaning six and the suffix "-ar".

Computation

Boogasix can be computed in the following process:

  • \(f(1)=6\)
  • \(f(2)=6^{6^{6^{6^{6^6}}}}\) with f(1) 6's, a.k.a. megafugasix
  • \(f(3)=6^{6^{\vdots^{6^6}}}\) with f(2) 6's
  • \(f(4)=6^{6^{\vdots^{6^6}}}\) with f(3) 6's
  • etc.
  • Boogasix equals \(f(f(f(f(f(6)))))\).

Approximations

Notation Approximation
Up-arrow notation \(6\uparrow\uparrow\uparrow\uparrow6\) (exact)
BEAF \(\{6,6,4\}\) (exact)
Hyper-E notation \(E(6)1\#1\#1\#6\) (exact)
Chained arrow notation \(6\rightarrow6\rightarrow4\) (exact)
Hyperfactorial array notation \(7!3\)
Fast-growing hierarchy \(f_5(5)\)
Hardy hierarchy \(H_{\omega^5}(5)\)
Slow-growing hierarchy \(g_{\varphi(3,0)}(6)\)

Sources

  1. Numbers from linear array notation | Steps Toward Infinity!
  2. Q-numbers - Beyond and beyond

See also

Numbers named with the fuga- family of prefixes

Gar-: garone · gartwo · garthree · garfour · garfive · garsix · garseven · gareight · garnine · garten · garhundred · gargoogol · gargoogolplex
Fz-: fzone · fztwo · fzthree · fzfour · fzfive · fzsix · fzseven · fzeight · fznine · fzten · fzeleven · fztwenty · fzhundred · fzgoogol · fzmillion · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgargantugoogolplex
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · fugagargantugoogolplex · fugagiggol
Megafuga-: megafugaone · megafugatwo · megafugathree · megafugafour · megafugafive · megafugasix · megafugaseven · megafugaeight · megafuganine · megafugaten · megafugaeleven · megafugatwelve · megafugathirteen · megafugafourteen · megafugafifteen · megafugasixteen · megafugaseventeen · megafugaeighteen · megafuganineteen · megafugatwenty · megafugathirty · megafugaforty · megafugafifty · megafugasixty · megafugaseventy · megafugaeighty · megafuganinety · megafugahundred · megafugagoogol · megafugagoogolplex · megafugagargoogolplex · megafugagargantugoogolplex
Booga-: boogaone · boogatwo · boogathree · boogafour · boogafive · boogasix · boogaseven · boogaeight · booganine · boogaten · boogahundred · boogagoogol · boogagoogolplex

Hyp cos' linear strong array notation numbers | exAN numbers
3-entry series
Tribo group: tribo · tetbo · pentbo · hexabo
Trientri group: trientri · tettro · pentro · hextro
Trientet group: triteto · trientet · penteto · hexteto
Trienpent group: tripeno · tetpeno · trienpent · hexpeno
Trienhex group: trihexo · tethexo · penhexo · trienhex