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Superfactorial

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The superfactorial is a factorial-based function with differing definitions.[1]

Pickover Edit

Clifford A. Pickover defines superfactorial as \(n\$ = {}^{(n!)}(n!) = \underbrace{n!^{n!^{n!^{.^{.^.}}}}}_{n!}\). (the factorial of n tetrated to itself or equivalently the factorial of n pentated to 2).

The above is also equal to \(n! \uparrow\uparrow n!\) or \(n! \uparrow\uparrow\uparrow 2\) in up-arrow notation

Using Hypercalc, Wolfram Alpha and bcalc, some values of Pickover's superfactorial are described below:

  • \(1$ = 1\)
  • \(2$ = 4\)
  • \(3$ = 10^{10^{10^{10^{36305.315801918918..}}}} = 4pt36305.315801918918.. = 5pt4.559970218821..\)
  • \(4$ = 24pt1.521987728335..\)
  • \(5$ = 120pt2.397626581446..\)
  • \(6$ = 720pt3.313389520154..\)
  • \(7$ = 5040pt4.270930686287..\)
  • \(8$ = 40320pt5.268800796659..\)
  • \(9$ = 362880pt.6.304819474820..\)
  • \(10$ = 362880pt7.376651198837..\)
  • \(11$ = 39916800pt8.482035348919..\)
  • \(12$ = 479001600pt9.618873548666..\)
  • \(13$ = 6227020801pt1.032830331015..\)
  • \(14$ = 87178291201pt1.078436584986..\)
  • \(15$ = 1307674368001pt1.120569877239..\)
  • \(...\)
  • \(100$ = (100!+1)pt2.204577320632..\)
  • \(1000$ = (1000!+1)pt3.410104470640..\)
  • \(10^6$ = (10^{6}!+1)pt6.745521015639..\)
  • \(\text{googol}$ = (googol!+2)pt2.008592123510..\)

Sloane and Plouffe Edit

Neil J.A. Sloane and Simon Plouffe define superfactorial as \(n\$ = \prod^{n}_{i = 1} i! = 1! \cdot 2! \cdot 3! \cdot 4! \cdot \ldots \cdot n!\), the product of the first \(n\) factorials. The first few values of \(n$\) for \(n = 1, 2, 3, \ldots\) are 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, 1834933472251084800000, 6658606584104736522240000000, 265790267296391946810949632000000000, 127313963299399416749559771247411200000000000, ... (OEIS A000178).

This superfactorial has an interesting relationship to the hyperfactorial: \(n\$ \cdot H(n) = n!^{n + 1}\). This may be proven by induction, with the base case \(1\$ \cdot H(1) = 1 = 1!^2\) and the following simple inductive step:

\begin{eqnarray} n\$ \cdot H(n) &=& n!^{n + 1} \\ n\$ \cdot H(n) \cdot (n + 1)! \cdot (n + 1)^{n + 1} &=& n!^{n + 1} \cdot (n + 1)! \cdot (n + 1)^{n + 1} \\ (n + 1)\$ \cdot H(n + 1) &=& (n + 1)!^{n + 2} \\ \end{eqnarray}

Daniel Corrêa Edit

In January 25th 2016 when editing this article, the Brazilian "amateur" googologist Daniel Corrêa aspired to create a new type of superfactorial.

The third definition for superfactorial (\(n\$\)), as proposed by Corrêa is:[2]

\(n\$ = (\underbrace{11...11}_{n}n)\times((\underbrace{11...11}_{n-1}n)!)\times((\underbrace{11...11}_{n-2}n)!^{2})\cdots((111n)!^{(n-3)})\times((11n)!^{(n-2)})\times(n!^{(n-1)})\\=\prod_{k=1}^{n}((10^k-1)\times \frac{n}{9})!^{n-k}\)

where \(!^{2}\), \(!^{(n-3)}\), \(!^{(n-2)}\) and \(!^{(n-1)}\) are from Nested factorial notation as defined by Aarex Tiaokhiao.

Considering the new function as described above, for the first three we have:

  • \(1\$ = 1\)
  • \(2\$ = 22\times2! = 22\times2 = 44\)
  • \(3\$ = 333\times33!\times3!^{2} = (333\times8683317618811886495518194401280000000\times720) \\ 3\$ = 2081912232286337906165442289650892800000000\)

Sources Edit

  1. Superfactorial from Wolfram MathWorld
  2. Uma nova função matemática!

See also Edit

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

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