Question about fast-growing functions
The sequence that produces Graham’s number is a well-known fast-growing sequence. I want to know how it contrasts with the following: Let F(x) be the function that produces the sequence and have it be defined for natural numbers n≥2. F(2)=2↑2=4 F(3)=3↑↑↑↑3=k F(4)=4↑(k)4=j F(5)=5↑(j)5... where ↑(x) denotes the number of arrows in an expression. Clearly this is similar to the G sequence. In fact, G(1)=F(3). Given that, is G(x)>F(x) for all positive integers? If not, around what size is the input that outputs a larger number in F's sequence given the same input to G's sequence?
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