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Thinking about superfactorials, especially related to that defined by Pickover, I thought why not try an extension to such functions? Pickover superfactorial is defined as \(n\$=\underbrace{n!^{n!^{n!^{.^{.^{.^{.}}}}}}}_{n!}\) that is the same of \(n![4]n!\). The idea of Greek factorials is to be an extension of the hyperoperation, exponent and base number of the superfactorial function. Initially I suggest eight Greek factorials as below:

  • \(n\alpha\$ = n![n+2]n! = n!\underbrace{\uparrow...\uparrow}_{n}n!\);
  • \(n\beta\$ = n![n!+2]n! = n!\underbrace{\uparrow...\uparrow}_{n!}n!\);
  • \(n\gamma\$ = n\$[n]n! = n\$\underbrace{\uparrow...\uparrow}_{n}n!\);
  • \(n\delta\$ = n![n]n\$ = n!\underbrace{\uparrow...\uparrow}_{n}n\$\);
  • \(n\theta\$ = n![n!]n\$ = n!\underbrace{\uparrow...\uparrow}_{n!}n\$\);
  • \(n\Sigma\$ = n\$[n]n\$ = n\$\underbrace{\uparrow...\uparrow}_{n}n\$\);
  • \(n\Phi\$ = n\$[n!]n\$ = n\$\underbrace{\uparrow...\uparrow}_{n!}n\$\);
  • \(n\Omega\$ = n\$[n\$]n\$ = n\$\underbrace{\uparrow...\uparrow}_{n\$}n\$\)

Greek factorials functions grows tremendously faster than superfactorial. The "slowest" one, \(n\alpha\$\), is much faster than Pickover superfactorial, see the examples of \(n\$\) and \(n\alpha\$\) for \(n = 1, 2\) and \(3\) below:

  • \(1\$ = 1\)
  • \(1\alpha\$ = 1\uparrow1 = 1\)
  • \(2\$ = 2!^{2!} = 2^2 = 4\)
  • \(2\alpha\$ = 2!\uparrow\uparrow2! = 4\uparrow\uparrow4 = 4^4 = 256\)
  • \(3\$ = 6\uparrow\uparrow6 = \underbrace{6\uparrow6\uparrow6\uparrow6\uparrow6\uparrow6}_{6} = 5pt4.559970..\), between gigafaxul and googoltriplexigong
  • \(3\alpha\$ = 6\uparrow\uparrow\uparrow6 = \underbrace{6\uparrow6\uparrow...\uparrow6\uparrow6}_{\underbrace{6\uparrow6\uparrow6\uparrow6\uparrow...\uparrow6\uparrow6\uparrow6\uparrow6}_{\underbrace{6\uparrow6\uparrow6\uparrow6\uparrow...\uparrow6\uparrow6\uparrow6\uparrow6}_{\underbrace{6\uparrow6\uparrow6\uparrow...\uparrow6\uparrow6\uparrow6}_{\underbrace{6\uparrow6\uparrow6\uparrow6\uparrow6\uparrow6}_{6}}}}}\), between quadgrand faxul and hexa-taxis

The strength of \(n\Omega\$\) function is very huge. As examples, \(2\Omega\$\) is equal to Tritet, \(\lbrace 4,4,4 \rbrace\) in BEAF or the fourth Ackermann number; \(3\Omega\$\) is \(\lbrace 3\$,3\$,3\$ \rbrace \) in BEAF, where \(3\$\) is \(6\uparrow\uparrow6\). In fact, the \(n\Omega\$\) function gives the \(n\$\)th Ackermann number as result, with \(n\$\) as Pickover superfactorial. Approximate notations for \(n\Omega\$\) are:

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