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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

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

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:

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