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Let me introduce the function based on factorial (!). I will call this function Super Staging Factorial Function. In this function we can go far beyond with the simple notation !. I gonna tell how I got this idea. I mainly got this inspiration by dysfunctionalfactorial's idea at http://www.artofproblemsolving.com/community/c1664h1003843 about factorial. This is not be confused with Faxul terms. The naming system focuses with the number 200, similar to Faxul terms. I use Faxul terms to see how fast this function grow perspective. I don't think this function would beat Expofaxul terms very fast. Numbers I have created is based on the name Lexol, meaning "Level Factorial", the first tier. And name Raxol, meaning "Rank Factorial", the second tier. I hasn't gotten to the third yet. We'll use pairs of vertical pipes (|) with number or chain (\(α\)) within those pairs of pipes, and a number in the subscript (\(β\)) outside the pipes, (AND if necessery, have a number superscript (\(γ\)) which describes the repeat of the function), and aswell as a input (\(x\)) to make those process:

\(|α|_β^γ(x)=output\)

Let's clearify some rules:

  • Rule 1: numbers in the function must be intergers that are 1 or greater.
  • Rule 2: \(|1|_1(x)=x!\)
  • Rule 3: \(|m|_n^1(x)=|m|_n(x)\)
  • Rule 4: \(|m|_n^{o+1} (x)=|m|_n^o(|m|_n^1(x))\)
  • Rule 5: \(|m|_{n+1} ^1(x)=|m|_n^x(x)\)
  • Rule 6: \(|m+1|_1(x)=|m|_x(x)\)
  • Rule 7: \(|...|1|...(\text{n+1 pairs of |'s})...|_1(x) = |...|x|...(\text{n pairs of |'s})...|_1(x)\)
    • Rule 1,3-6 is also applied on multiple |'s
  • Rule 8: \(|...|m|... (\text{n pairs of |'s}) ...|_1(x) = |n,m|_1(x)\)
  • Rule 9: \(|m+1,1_1,1_2,...,1_(n+1) |_1(x)=|m,x,1_1,1_2,...,1_n|_1(x)\)

Examples:

  • \(|1|_1(3)=3!=3·2·1=6\)
  • \(|1|_1(10)=10!=3.628.800\)
  • \(|1|_1(100)=100!≈9,33262154439·10^{157}\)
  • \(|1|_1(200)=200!≈7,9·10^{375}\), I name this number Lexol. Also equal to Faxul.
  • \(|1|_1(|1|_1(3))=(3!)!=720\), this can be compressed into \(|1|_1^2(3)\).
  • \(|1|_1^2(10)=(10!)!≈5,2279·10^{22228109}\)
  • \(|1|_1^2(200)=(200!)!≈10^{9,48539·10^378} \), which is named Kilolexol.

More names:

  • Megalexol
  • Gigalexol
  • Dialexol
  • Trialexol
  • Grand Lexol
  • Bigrand Lexol
  • Ultimate Lexol
  • Uldimate Lexol
  • Ulquintimega Quadgrand Exatetralexol

Further notations:

  • Raxol = \(|1|1|_1(200) = |200_1,200_2,...,200_{200}|_1(200)\)
  • Ultimate Raxol


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