7 factorials you probably didn't know

Here are 7 less common factorials that you probably didn't know: double factorial, subfactorial, primorial, super factorial, exponential factorial, hyper factorial

0:00 new factorials?
0:53 double factorial
3:11 subfactorial
5:33 primorial
7:13 super factorial by Sloane
8:38 super factorial by Pickover
10:22 exponential factorial
11:28 hyper factorial

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[Mephisto]

If that is the Hyper factorial, the Pickover factorial should be named UltraMegaBlaster factorial instead of merely Super.

[luggepytt]

Two observations:

1. The double factorial is also known as the semifactorial, which I personally think makes more sense, since you are only multiplying half of the numbers less than or equal to n.

2. All this super-duper-mega-hyper factorial stuff reminds me of when we were kids, and got into an argument about things like whose car was faster, or whose daddy earned more money, like little boys often do. It usually went something like this:

- A hundred.
- Two hundred.
- A thousand!
- A thousand thousand!
- Ten times more than you can say!!!

(And no, that's not a triple factorial. It's just three exclamation marks.)

[erik]

My TI-nspire sadly passed away calculating the 24 Power Tower... Rest in Pieces

[digitig]

"Five times three times one. You can do that by yourself."

*Finally* he gets to a level of mathematics I can do!

[neilgerace]

"Multiple exclamation marks are a sure sign of a diseased mind."

Sir Terry Pratchett

[lorenzohsu]

The real question is: "How do you seamlessly switch between pens?!"

[PhantomKING]

For the primorial, ig 1# = 1 makes the most sense to me.

Ways to arrive at this conclusion:
1: You also multiply by 1 even if it isn't a prime.
2: Since 2 is a prime number, (2–1)# must be 2#/2, which in this case is 1.
3: An empty multiplication is 1.

[MagnusSkiptonLLC]

10:14 I calculated it, but YouTube doesn't allow posting comments so large they physically create black holes in the server. I've submitted a bug report, when it's fixed I'll get back to you.

[Arthur]

Usually the empty product is defined as 1 and empty sum as 0. So if the set of primes equal or lower than 1 is empty the product should be 1 by convention

[Hepad_]

The exponential factorial should use the euro (€) symbol. It's still a monetary symbol so it would remind us of the dollar symbol, and it symbolizes a E, just like 'exponential'.

[angel-ig]

6:46 There's no primes less than or equal to one. Therefore, the solution is the product of the empty set, which is 1: the multiplicative identity.

[GvinahGui]

It's funny how the Hyper factorial gives way smaller numbers then the Super factorials (Pickover)

[TheRailfanner]

It's a shame how most math students are never introduced to the double factorial and/or subfactorial during Calc 2. I feel that knowing these concepts would make comprehending series a little easier.

[stevemonkey]

I am surprised no one has come up with a Super Hyper Factorial

[KingGrio]

In the end my takeaway is:
-the first 3 are useful notations
-number 5 allows to write the biggest numbers with only few symbols
-I don't see what 4 is good for but I have a feeling I could run into it naturally
-I don't see what 6 is good for and have no idea when I'll ever need it
-7 is bigger than 4

[deandelvin]

Do mathematicians secretly hate humanity

[AliKhanMaths]

Wow this is pretty fascinating - I didn't know some these existed, and their uses are also interesting! Videos like yours inspire me to share my own maths content as well!

[ffggddss]

Some crazy stuff! And some not-so-crazy.
I chuckled silently when you asked for calculator help with the power-tower, 24^(24^(24^(...^24)...)).
I was picturing some poor cuss actually trying to work this out on a calculator. Even taking the log will only "reduce" the tower by 1 "level."
And you didn't even crack a smile when you said that.

Incidentally, I would say that 1# = 1, because it's a vacuous product – there are no primes ≤ 1.

Fred

[giovannicaiolo]

7:06
I don’t know the actual answer but I would guess 1#=1 for a reason similar to why 0!=1
We can define (n+1)# as =n# if n+1 isn’t prime and =(n+1) x n# if n+1 is prime
2 is prime and we know that 2#=2 so 2#=2=2 x 1# so 1#=1

[lego]

It's crazy that the number of derangements !n == the closest integer to n! / e. We looked at the formula for derangements on the first day of my combinatorics lecture because the formula was so cool.