User:Cloudy176/Department of bubbly negative numberbottles/Cell function

The cell function is a failed function created by SpongeTechX.


 * Content below was originally on Cell function; major contributors were SpongeTechX and Googleaarex. The content was edited to fit my style.
 * See also: Talk:Cell function

The cell function is a googological function invented by the googology wiki user, SpongeTechX. Let us iterate them with a "∷" symbol. It uses all of the same numbers to keep it simple. Let's start putting out some examples.


 * 1∷ = 1
 * 2∷ = 2 + 2 = 4
 * 3∷ = 3 + 3 x 3 = 12
 * 4∷ = 4 + 4 x 4 ↑ 4 = 516
 * 5∷ = 5 + 5 x 5 ↑ 5 ↑↑↑↑↑ 5
 * 6∷ = 6 + 6 x 6 ↑ 6 ↑↑↑↑↑↑ 6 ↑6+1 6
 * 7∷ = 7 + 7 x 7 ↑ 7 ↑7 7 ↑7+2 7 ↑(7 ↑ 7+2 7) 7
 * 8∷ = 8 + 8 x 8 ↑ 8 ↑8 8 ↑8+3 8 ↑(8 ↑ 8+3 8) 8 ↑(8 ↑ (8 ↑ 8+3 8) 8) 8

9∷, 10∷, etc. We can also define nested cells. (n∷)∷ = n∷2, (n∷)∷)∷ = n∷3, etc. This function is very similar to mixed factorials, except much stronger. We can pronounce these cells as "cell one", "cell two", "cell three" or "cell of one", "cell of two", "cell of three", or "1st cell", "2nd cell", 3rd cell".

Extension

 * This section was commented out in the original, due to being not apparent in source

An extension can be iterated as n∷(x = n∷n∷n∷ . . . n∷n∷ (with x n∷'s). Then n∷((x = n::(n::...(n::(n:: (with x n::'s), n∷(((x = n::((n::...((n::((n:: (with x n::'s), etc.

Also n::(kx = n::((...((x (with k ('s), n::((kx = n::(...n::( n::(xx... x (with k nested), n::(((kx = n::((...n::( (n::(xx... x (with k nested), etc.

Next, n::(( k 2 k x = n::(((...((kx (k2 ('s), n::(( ( k 3 k2 k x = n::(( ((...((k 2 k x (k3 ('s), etc.

Numbers defined using cell function

 * Dekacell = 10∷
 * Duo-dekacell = 10∷2