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

Pickover

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

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

  • \(1$ = 1\)
  • \(2$ = 4\)
  • \(3$ = 10^{10^{10^{10^{36305.315801918918..}}}} = 4pt36305.315801918918.. = 5pt4.559970218821..\)
  • \(4$ = 24pt1.521987728335..\)
  • \(5$ = 120pt2.397626581446..\)
  • \(6$ = 720pt3.313389520154..\)
  • \(7$ = 5040pt4.270930686287..\)
  • \(8$ = 40320pt5.268800796659..\)
  • \(9$ = 362880pt6.304819474820..\)
  • \(10$ = 3628800pt7.376651198837..\)
  • \(11$ = 39916800pt8.482035348919..\)
  • \(12$ = 479001600pt9.618873548666..\)
  • \(13$ = 6227020801pt1.032830331015..\)
  • \(14$ = 87178291201pt1.078436584986..\)
  • \(15$ = 1307674368001pt1.120569877239..\)
  • \(...\)
  • \(100$ = (100!+1)pt2.204577320632..\)
  • \(1,000$ = (1,000!+1)pt3.410104470640..\)
  • \(1,000,000$ = (1,000,000!+1)pt6.745521015639..\)
  • \(\text{googol}$ = (googol!+2)pt2.008592123510..\)

Sloane and Plouffe

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, 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

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:

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

Sources

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

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
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial