Automatic Set

A -automatic set is a set of integers whose base- representations form a regular language, i.e., a language accepted by a finite automaton or state machine. If bases  and  are incompatible (do not have a common power) and if an -automatic set  and -automatic set  are both of density 0 over the integers, then it is believed that  is finite. However, this problem has not been settled.

Some automatic sets, such as the 2-automatic consisting of numbers whose binary representations contain at most two 1s: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, ... (OEIS A048645) have a simple arithmetic expression. However, this is not the case for general -automatic sets.

REFERENCES:

Cobham, A. "On the Base-Dependence of Sets of Numbers Recognizable by Finite Automata." Math. Systems Th. 3, 186-192, 1969.

Cobham, A. "Uniform Tag Sequences." Math. Systems Th. 6, 164-192, 1972.

Sloane, N. J. A. Sequence A048645 in "The On-Line Encyclopedia of Integer Sequences."