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BinaryIteratedLog

is the inverse of binary tetration (repeated exponentiation):

[◼]	BinaryIteratedLog

[2^2^2^2]

In[]:=

4

Out[]=

A slightly larger input shows a step-like jump in the value of BinaryIteratedLog:

[◼]	BinaryIteratedLog

[2^2^2^2+1]

In[]:=

5

Out[]=

Make a table of the binary iterated logarithm of the first 30 integers:

Table

[◼]	BinaryIteratedLog

[i],{i,0,30}

In[]:=

{0,0,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4}

Out[]=