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.
See also
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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."
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Automatic Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AutomaticSet.html