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Hyper-Moser notation
TypeHierarchy
Growth rate$$f_{\omega^\omega}(n)$$

Hyper-Moser notation is an extension of Steinhaus-Moser notation invented by Aarex Tiaokhiao.[1] Formally it is defined as follows:

• Let $$\#$$ denote the rest of the tuple of arguments.
• $$M(n,m \#) = \underbrace{M(M(M(...,m-1\#),m-1\#),m-1\#)}_{n~M\text s}$$ ($$m>1$$)
• $$M(n,1) = n^n$$ (only 2 entries and $$m=1$$), or n in a triangle = triangle(n).
• $$M(\#\ 0) = M(\#)$$ (last entry is 0)
• $$M(n,0,0,...,0,0,m) = M(\underbrace{n,n...n,n}_{n+1},m-1)$$ (otherwise)

### Etymology

The M stands for Moser.

### Examples

$$M(n,2) = n$$ in a square

$$M(n,3) = n$$ in a circle or pentagon

$$M(2,3) =$$ Mega

$$M(n,m) = n$$ in a (2+m)-gon

$$M(2,M(2,3)-2) =$$ Moser

$$M(n,0,1) = M(n,n)$$

$$M(2,1,1) = M(M(2,0,1),0,1) = M(M(2,2),0,1) = M(256,256)$$

$$M(3,1,1) = M(M(M(3,0,1),0,1),0,1) = M(M(M(3,3),0,1),0,1) = M(M(M(3,3),M(3,3)),0,1) = M(M(M(3,3),M(3,3)),M(M(3,3),M(3,3)))$$

$$M(65,1,1) >$$ Graham's Number

$$M(2,2,1) = M(M(256,256),1,1)$$

$$M(3,2,1) = M(M(M(M(M(3,3),M(3,3)),M(M(3,3),M(3,3))),1,1),1,1)$$

$$M(2,3,1) = M(M(2,2,1),2,1)$$

$$M(n,0,2) = M(n,n,1)$$

### Sources

1. Tiaokhiao, AarexHyper-Moser notation. Retrieved 2013-03-30.
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