Factorials, Hyperfactorials and Superfactorials

Jai Ganesh · 2007-02-20 02:37:41

The factorial notation must be familiar to most of you.

n! (read as n factorial) is defined as
n!=n(n-1)(n-2)(n-3)...........4 x 3 x 2 x 1.
Thus, 2!=2 x 1 = 2
3! = 3 x 2 x 1 = 6,
4! = 4 x 3 x 2 x 1 = 24
5! = 5 x 4 x 3 x 2 x 1 = 120
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 and so on.
Facotrials are useful in Combinatorics (Permutations, Combinations etc.), Probability theory, Binomial theorem, Calculus etc.

Hyperfactorial is defined as

Thus,
H(1) = 1,
H(2) = 4,
H(3) = 108 and so on.

Finally, the Superfactorial.

Clifford Pickover in his 1995 book Keys to Infinity defined the superfactorial of n as

When expressed in Knuth's up-arrow notation.

n$=n!^^n!

For example,

The function grows very rapidly and as n increases, the tower of powers increases at a very quick rate.

100$ would have more powers in the tower than a Googol! And
1000$ would have more powers in the tower than

.
These are extremely large numbers, and absolutely useless to a common man!
That is because a person may never encounter a number greater than

for most of his/her life, and certainly never ever think of anything near

unless he's/she's a mathematician!

Patrick · 2007-02-20 04:58:25

Is the hyperfactorial meant to be:

? If it isn't, then I'm not sure I understand your notation

Jai Ganesh · 2007-02-20 16:09:46

Yes, you are correct, Patrick!
That is what it is.
I had even given examples of H(1)=1,
H(2)=4, H(3)=1.2².3³=1 x 4 x 27=108.

Toast · 2007-02-20 19:12:02

In Knuth's upper arrow notation, how many powers do you raise n to?

Patrick · 2007-02-20 19:45:38

ganesh wrote:

Yes, you are correct, Patrick!
That is what it is.
I had even given examples of H(1)=1,
H(2)=4, H(3)=1.2².3³=1 x 4 x 27=108.

Oh well, I guess I'm turning blind.. Thanks for sharing the knowledge though Had never heard of hyperfactorials before!

Zhylliolom · 2007-02-20 20:16:43

Toast wrote:

In Knuth's upper arrow notation, how many powers do you raise n to?

means raise a to itself n-1 times. For example,

Then basically there is a "tower" of n a's.

I can post more on this notation once I figure out how to make the LaTeX work here. (does someone know how to do underbraces in this forum? I can't see you get it to work)

Patrick · 2007-02-21 09:15:09

Zhylliolom wrote:

Toast wrote:
In Knuth's upper arrow notation, how many powers do you raise n to?
means raise a to itself n-1 times. For example,
Then basically there is a "tower" of n a's.
I can post more on this notation once I figure out how to make the LaTeX work here. (does someone know how to do underbraces in this forum? I can't see you get it to work)

Last edited by Patrick (2007-02-21 09:15:31)

Jai Ganesh · 2007-02-21 16:28:21

Toast,
This page gives details of Knuth's up-arrow notation. The operation becomes much more complicated when the number of up-arrows is more.

iamaditya · 2016-12-03 22:34:58

Neil Sloane and Simon Plouffe defined a superfactorial in The Encyclopedia of Integer Sequences (Academic Press, 1995) to be the product of the first n factorials. So the superfactorial of 4 is
sf(4)=4!*3!*2!*1!=288 and
sf(n)=n!*(n-1)!*(n-2)!..........2!*1!

Alternative definition

Clifford Pickover in his 1995 book Keys to Infinity used a new notation, n$, to define the superfactorial
x$=(x!)^(x!)^(x!)^(x!).....{x! times}

Source:Wikipedia

NakulG · 2016-12-13 04:33:27

Where are these used? In Physics?

bobbym · 2016-12-13 05:23:55

Number theory maybe.

Agnishom · 2016-12-13 15:52:47

In Knuth and Graham's book, Concrete Mathematics, the authors use notations like falling factorials for discrete mathematics.

bobbym · 2016-12-13 17:37:40

That is true since Knuth invented the arrow notation.

Math Is Fun Forum

#1 2007-02-20 02:37:41

Factorials, Hyperfactorials and Superfactorials

#2 2007-02-20 04:58:25

Re: Factorials, Hyperfactorials and Superfactorials

#3 2007-02-20 16:09:46

Re: Factorials, Hyperfactorials and Superfactorials

#4 2007-02-20 19:12:02

Re: Factorials, Hyperfactorials and Superfactorials

#5 2007-02-20 19:45:38

Re: Factorials, Hyperfactorials and Superfactorials

#6 2007-02-20 20:16:43

Re: Factorials, Hyperfactorials and Superfactorials

#7 2007-02-21 09:15:09

Re: Factorials, Hyperfactorials and Superfactorials

#8 2007-02-21 16:28:21

Re: Factorials, Hyperfactorials and Superfactorials

#9 2016-12-03 22:34:58

Re: Factorials, Hyperfactorials and Superfactorials

#10 2016-12-13 04:33:27

Re: Factorials, Hyperfactorials and Superfactorials

#11 2016-12-13 05:23:55

Re: Factorials, Hyperfactorials and Superfactorials

#12 2016-12-13 15:52:47

Re: Factorials, Hyperfactorials and Superfactorials

#13 2016-12-13 17:37:40

Re: Factorials, Hyperfactorials and Superfactorials

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