Relationships Between the Busy Beaver Function and Mathematics I: What is the Smallest n such that BB(n) > ... ?

1 This is the first of (hopefully only) two papers In this paper we ask the following question: for a variety of computable functions f, what is the smallest n such that BB(n) > f(n). In the sequel, we will ask questions such as: What is the smallest Turing machine M such that Goldbach's conjecture is true iff M never halts? In both papers all concepts and definitions are due to others unless otherwise noted. For example, if I wrote We define a hierarchy similar to the Wainer hierarchy the definition we give is already known.