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

Pickover[]

Superfactorial (Pickover)
Notation\(n$\)
TypeCombinatorial
Based onFactorial
Growth rate\(f_{3}(n)\)
AuthorPickover
Year1995


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) in his book Keys to Infinity.

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

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

  • \(1$ = 1\)
  • \(2$ = 4\)
  • \(3$ = 10^{10^{10^{10^{36305.315801918918..}}}} = ...9127238656\)
  • \(4$ \approx E1.521987728335\#24\) (Hyper-E notation)
  • \(5$ \approx E2.397626581446\#120\)
  • \(6$ \approx E3.313389520154\#720\)
  • \(7$ \approx E4.270930686287\#5040\)
  • \(8$ \approx E5.268800796659\#4032\)
  • \(9$ \approx E6.304819474820\#362880\)
  • \(10$ \approx E7.376651198837\#3628800\)
  • \(11$ \approx E8.482035348919\#39916800\)
  • \(12$ \approx E9.618873548666\#479001600\)
  • \(13$ \approx E1.032830331015\#6227020801\)
  • \(14$ \approx E1.078436584986\#87178291201\)
  • \(15$ \approx E1.120569877239\#1307674368001\)
  • \(...\)
  • \(100$ \approx E2.204577320632\#(100!+1)\)
  • \(1,000$ \approx E3.410104470640\#(1,000!+1)\)
  • \(1,000,000$ \approx E6.745521015639\#(1,000,000!+1)\)
  • \(\text{googol}$ \approx E2.008592123510\#(\text{googol}!+2)\)
  • \(\text{googolplex}$ \approx E3.727193826\#(\text{googolplex}!+2)\)

Sloane and Plouffe[]

Superfactorial (Sloane and Plouffe)
Notation\(n$\)
TypeCombinatorial
Based onFactorial
Growth rate\(f_{2}(n)\)
AuthorSloane and Plouffe
Year1995

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 \(n = 0,1, 2, 3, \ldots\) are 1, 1, 2, 12, 288, 34,560, 24,883,200, 125,411,328,000, 5,056,584,744,960,000, 1,834,933,472,251,084,800,000, 6,658,606,584,104,736,522,240,000,000, 26,579,026,7296,391,946,810,949,632,000,000,000, 127,313,963,299,399,416,749,559,771,247,411,200,000,000,000, ... (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}

Specific numbers[]

  • 288 is the fourth superfactorial number.
    • It is also the sum of the self-powers of the first four positive numbers.
    • Furthermore, it is equal to 16!!!!!!!.
    • Since samarium-146 and plutonium-244 used to be regarded as primordial nuclides, some sources list 288 primordial nuclides.

Daniel Corrêa[]

Superfactorial (Daniel Corrêa)
Notation\(n$\)
TypeCombinatorial
Based onFactorial
Growth rate\(f_{3}(n)\)
AuthorDaniel Corrêa
Year2016

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:

  • \(1\$ = 1\)
  • \(2\$ = 22\times2! = 22\times2 = 44\)
  • \(3\$ = 333\times33!\times3!^{2} = (333\times8,683,317,618,811,886,495,518,194,401,280,000,000\times720) \\ = 2,081,912,232,286,337,906,165,442,289,650,892,800,000,000\)

Sources[]

See also[]

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 · Primorial · Compositorial · Semiprimorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Nested Factorials: Tetorial · Petorial · Ectorial · Zettorial · Yottorial
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial
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