Superfactorial

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

Pickover[]

Superfactorial (Pickover)
Notation	$n$$
Type	Combinatorial
Based on	Factorial
Growth rate	$f_{3}(n)$
Author	Pickover
Year	1995

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$$
Type	Combinatorial
Based on	Factorial
Growth rate	$f_{2}(n)$
Author	Sloane and Plouffe
Year	1995

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.

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.^[2] \[G(1+z)=(2\pi)^{z/2} \exp\left(- \frac{z+z^2(1+\gamma)}{2} \right) \, \prod_{k=1}^\infty \left\{ \left(1+\frac{z}{k}\right)^k \exp\left(\frac{z^2}{2k}-z\right) \right\}\] where $\, \gamma$ is the Euler–Mascheroni constant, and $\prod$ denotes multiplication.

Daniel Corrêa[]

Superfactorial (Daniel Corrêa)
Notation	$n$$
Type	Combinatorial
Based on	Factorial
Growth rate	$f_{3}(n)$
Author	Daniel Corrêa
Year	2016

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:^[3]

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

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[]

↑ Superfactorial from Wolfram MathWorld
↑ E. W. Barnes, "The theory of the G-function", Quarterly Journ. Pure and Appl. Math. 31 (1900), 264–314.
↑ Uma nova função matemática!