A lower bound RADO's sigma function for binary turing machines

In this note we show how to construct some simply-configured N-state binary Turing machines that will start on a blank tape and eventually halt after printing a very large number of ones. The number of ones produced by these machines can be expressed analytically in terms of functional difference equation. The latter expression furnishes the best lower bound presently known for Rado's noncomputable function, Σ(N), when N ≫ 5.