The thinnest sequence which contains 1, and whenever it contains , also contains , , and : 1, 2, 4, 5, 8, 9, 10, 14, 15, 16, 17, ... (Sloane's A005658).
References
Guy, R. K. ``Klarner-Rado Sequences.'' §E36 in Unsolved Problems in Number Theory, 2nd ed.
New York: Springer-Verlag, p. 237, 1994.
Klarner, D. A. and Rado, R. ``Linear Combinations of Sets of Consecutive Integers.'' Amer. Math. Monthly 80, 985-989, 1973.
Sloane, N. J. A. Sequence
A005658/M0969
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.