Visualization of hextro using exponentiation

Hextro is equal to s(6,6,3) in strong array notation.[1] The term was coined by wiki user Hyp cos. It can be represented in up-arrow notation as \(6\uparrow\uparrow\uparrow6\) (6 pentated to 6).

This number is also equal to s(6,2,4) and s(6,3,1,2). It is approximately E36305#4#5 in Hyper-E notation.

Etymology

The name of this number is based on the word "hexa" and "tri" (Greek for 6 and 3), meaning a base-6 number, generated using hyper-3 operation based on exponentiation.

Computation

Hextro can be computed in the following process:

  • \(f(1)=6\)
  • \(f(2)=6^6=46656\)
  • \(f(3)=6^{6^6}\approx2.6591\times10^{36305}\)
  • \(f(4)=6^{6^{6^6}}\)
  • etc.
  • Hextro equals \(f(f(f(f(f(6)))))\).

Approximations

Notation Approximation
Arrow notation \(6\uparrow\uparrow\uparrow6\) (exact)
BEAF \(\{6,6,3\}\) (exact)
Steinhaus-Moser Notation 6[5]
Hyper-E notation \(\textrm E[6]1\#1\#6\) (exact)
Chained arrow notation \(6\rightarrow6\rightarrow3\) (exact)
Hyperfactorial array notation \(7!2\)
Fast-growing hierarchy \(f_4(5)\)
Hardy hierarchy \(H_{\omega^4}(5)\)
Slow-growing hierarchy \(g_{\zeta_0}(6)\)

See also

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

Sources

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