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Hyperfactorial array notation

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Hyperfactorial array notation
Type Multidimensional
Based on factorial

Hyperfactorial array notation is a large number notation invented by Lawrence Hollom.[1] It was first developed in April 2013.

Basics Edit

Now we need to define arrays. Each array consists of a finite sequence of zero or more entries. Each entry consists of either a positive integer or another array (and these arrays can only nest finitely). An example of a valid array is

[1,1,[1,2,[3],4,[],1,1],1,3,10,1,[4,[4,3,1],5,6],1,[1,2],1,1]

First, we define the following notation:

\(n!m = n\uparrow^{m}(n-1)\uparrow^{m}(n-2)\cdots n\uparrow^{m} 3 \uparrow^{m} 2 \uparrow^{m} 1\)

Hyperfactorial array notation defines a function \(n!A\), where \(A\) is an array. An example of a well-formed expression in hyperfactorial array notation is \(5![6, [7, 8], 9]\).

Linear arrays Edit

Define the active entry as the first entry in the array that is not 1. This is analogous to BEAF's pilot.

  • Any ones may be cropped off the end of an array:
    \([@, 1] = [@]\)
  • Any empty array can simply be replaced with n:
    \([] = n\)
  • If the first entry is a number \(k>1\):
    \(f(a) = a![k-1,@]\)
    \(n![k,@] = f^n(n)\)
  • Otherwise:
    \(n![1,1,\cdots,1,1,[[...[[k @]]...]],@] = n![1,1,\cdots,1,[1,1,\cdots,1,1,[[...[[1 @]]...]],@],[[...[[k-1 @]]...]],@]\)

Here \(@\) indicates the rest of the array.

Analysis Edit

HAN is quite new compared to other array notations, and its growth rate has not yet been agreed on. Hollom believes that it reaches all the way to the Takeuti-Feferman-Buchholz ordinal.

Sources Edit

  1. Hollom, Lawrence. Hyperfactorial array notation. Retrieved 2013-06-08.

External links Edit

See also Edit


Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Others: Alternating factorial · Hyperfactorial · Primorial · q-factorial · Roman factorial · Subfactorial · Superfactorial by Simon and Plouffe · Torian · Factorexation · Mixed Factorial


Hyperfactorial array notation numbers: Minor Faxul Group | Major (Giaxul)

Faxul group: Faxul · Kilofaxul · Megafaxul · Gigafaxul · Terafaxul · Petafaxul · Exafaxul · Grand Faxul · Grand Kilofaxul · Grand Megafaxul · Grand Gigafaxul · Grand Terafaxul · Grand Petafaxul · Grand Exafaxul · Bigrand Faxul · Bigrand Kilofaxul · Bigrand Megafaxul · Trigrand Faxul · Trigrand Kilofaxul · Trigrand Megafaxul · Quadgrand Faxul · Quintgrand Faxul
Expofaxul group: Expofaxul · Kiloexpofaxul · Megaexpofaxul · Gigaexpofaxul · Teraexpofaxul · Petaexpofaxul · Exaexpofaxul · Grand expofaxul · Grand kiloexpofaxul · Grand megaexpofaxul · Grand gigaexpofaxul · Grand teraexpofaxul · Grand petaexpofaxul · Grand exaexpofaxul · Bigrand expofaxul · Bigrand kiloexpofaxul · Bigrand megaexpofaxul · Trigrand expofaxul · Trigrand kiloexpofaxul · Trigrand megaexpofaxul · Quadgrand expofaxul · Quintgrand expofaxul
Tetrofaxul group: Tetrofaxul · Kilotetrofaxul · Megatetrofaxul · Gigatetrofaxul · Teratetrofaxul · Petatetrofaxul · Exatetrofaxul · Grand tetrofaxul · Grand kilotetrofaxul · Grand megatetrofaxul · Grand gigatetrofaxul · Grand teratetrofaxul · Grand petatetrofaxul · Grand exatetrofaxul · Bigrand tetrofaxul · Bigrand kilotetrofaxul · Bigrand megatetrofaxul · Trigrand tetrofaxul · Trigrand kilotetrofaxul · Trigrand megatetrofaxul · Quadgrand tetrofaxul · Quintgrand tetrofaxul
Pentofaxul group: Pentofaxul · Kilopentofaxul · Megapentofaxul · Gigapentofaxul · Grand pentofaxul · Grand kilopentofaxul · Grand megapentofaxul · Grand gigapentofaxul · Bigrand pentofaxul · Trigrand pentofaxul
Hexofaxul group: Hexofaxul · Kilohexofaxul · Megahexofaxul · Gigahexofaxul · Grand hexofaxul · Grand kilohexofaxul · Grand megahexofaxul · Grand gigahexofaxul · Bigrand hexofaxul · Trigrand hexofaxul
Hyperfaxul group: Hyperfaxul · Kilohyperfaxul · Megahyperfaxul · Gigahyperfaxul · Terahyperfaxul · Petahyperfaxul · Exahyperfaxul · Grand hyperfaxul · Grand kilohyperfaxul · Grand megahyperfaxul · Grand gigahyperfaxul · Grand terahyperfaxul · Grand petahyperfaxul · Grand exahyperfaxul · Bigrand hyperfaxul · Bigrand kilohyperfaxul · Bigrand megahyperfaxul · Trigrand hyperfaxul · Trigrand kilohyperfaxul · Trigrand megahyperfaxul · Quadgrand hyperfaxul · Quintgrand hyperfaxul

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