Visualization of boogasix using exponentiation
Boogasix is equal to \(6\uparrow\uparrow\uparrow\uparrow6\), or six hexated to 6.[1]
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.[2]
Boboris02 calls this number Hexar, which also equals \(Q_{1,0}(6)=Q_{6}(6)\) in the Q-supersystem.[3]
Googology Wiki user HaydenTheGoogologist2009 calls this number Hayden's Trihex, and it's equal to 6(6)6 in Hayden's Array Notation.[4]
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 |
|---|---|
| Arrow notation | \(6\uparrow\uparrow\uparrow\uparrow6\) (exact) |
| Bowers' Exploding Array Function | \(\{6,6,4\}\) (exact) |
| Bird's array notation | \(\{6,6,4\}\) (exact) |
| Chained arrow notation | \(6\rightarrow6\rightarrow4\) (exact) |
| Hyper-E notation | \(E(6)1\#1\#1\#6\) (exact), \(E36305.31580191892\#4\#5\#5\) |
| Hyperfactorial array notation | \(7!3\) |
| Steinhaus-Moser Notation | 6[6] |
| Strong array notation | s(6, 6, 4) (exact) |
| X-Sequence Hyper-Exponential Notation | 6{4}6, 6{X}4 (both are exact) |
| Fast-growing hierarchy | \(f_5(5)\) |
| Hardy hierarchy | \(H_{\omega^5}(5)\) |
| Slow-growing hierarchy | \(g_{\varphi(3,0)}(6)\) |
Sources[]
- ↑ 2.5A. Googological prefixes and suffixes Retrieved 2021-07-22.
- ↑ Numbers from linear array notation | Steps Toward Infinity!
- ↑ Q-numbers - Beyond and beyond
- ↑ Hayden's Big Numbers. Retrieved 2022-04-20.
See also[]
Gar-: garone · gartwo · garthree · garfour · garfive · garsix · garseven · gareight · garnine · garten · garhundred · garmillion · gargoogol · gareceton · gartrialogue · gargoogolplex · gargiggol
Fz-: fzone · fztwo · fzthree · fzfour · fzfive · fzsix · fzseven · fzeight · fznine · fzten · fztwenty · fzthirty · fzhundred · fzthousand · fzmillion · fzgoogol · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgiggol
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · fugagiggol
Megafuga-: megafugaone · megafugatwo · megafugathree · megafugafour · megafugafive · megafugasix · megafugaseven · megafugaeight · megafuganine · megafugaten · megafugahundred · megafugagoogol · megafugagoogolplex · megafugagargoogolplex · megafugagargantugoogolplex · megafugagrangol
Extensions:
Booga-:: boogaone · boogatwo · boogathree · boogafour · boogafive · boogasix · boogaten · boogahundred · boogagoogol · boogagoogolplex
Gag-:: gagone · gagtwo · gagthree · gagfour · gagfive · gagsix · gagseven · gageight · gagnine · gagten · gaggoogol · gaggoogolplex
Tetentri group: duprimitol · duprimitolplex · duprimibol · tetentri
Tetentet group: truprimitol · truprimitolplex · truprimibol · truprimibolplex · tetentet
Tetenpent group: quadprimitol · quadprimitolplex · quadprimibol · quadprimibolplex · tetenpent
Tetenhex group: quinprimitol · quinprimitolplex · quinprimibol · quinprimibolplex · tetenhex
Chainprimol group: chainprimol · chainprimolplex · chainpribol · chainpribolplex · chainduprimol · chainduprimolplex · chaindupribol · chaindupribolplex · chaintruprimol · chaintruprimolplex · chaintrupribol · chaintrupribolplex
Pententri group: duchainol · duchainolplex · duchainbol · duchainbolplex · duchainprimol · duchainprimolplex · duchainpribol · duchainpribolplex · duchainduprimol · duchainduprimolplex · duchaindupribol · pententri
Pententet group: truchainol · truchainolplex · truchainbol · truchainbolplex · truchainprimol · truchainprimolplex · truchainpribol · truchainpribolplex · truchainduprimol · truchainduprimolplex · truchaindupribol · truchaindupribolplex · pententet · pentenpent · pentenhex
Choichainol group: choichainol · choichainolplex · choichainbolplex · choichainpribolplex · choichaindupribolplex · choiduchaindupribolplex
Hexentri group: hexentri · hexentet · hexenpent · hexenhex