Superfactorial

The superfactorial is a factorial-based function with differing definitions.Superfactorial from Wolfram MathWorld
== 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  or 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 ==
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, ... ().
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 -146 and plutonium-244 used to be regarded as primordial nuclides, some sources list 288 primordial nuclides.
=== Extension to complex number ===
Barnes G-function, defined as follows, can be regarded as an extension of Sloane and Plouffe version of superfactorial to continuous function on complex number.E. W. Barnes, "The theory of the G-function", Quarterly Journ. Pure and Appl. Math. 31 (1900), 264–314.
\[G(1+z)=(2\pi)^{z/2} \exp\left(- \frac{z+z^2(1+\gamma)}{2} \right) \, \prod_{k=1}^\infty \left\{ \left(1+\frac{z}{k}\right)^k \exp\left(\frac{z^2}{2k}-z\right) \right\}\]
where \(\, \gamma\) is the , and \(\prod\) denotes multiplication.
== 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:Uma nova função matemática!
\(n\$ = (\underbrace{11...11}_{n}n)\times((\underbrace{11...11}_{n-1}n)!)\times((\underbrace{11...11}_{n-2}n)!^{2})\cdots((111n)!^{(n-3)})\times((11n)!^{(n-2)})\times(n!^{(n-1)})\\=\prod_{k=1}^{n}((10^k-1)\times \frac{n}{9})!^{n-k}\)
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 ==
ja:超階乗
Category:Factorials
Category:Functions