## FANDOM

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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: