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The set of numbers generated by excluding the Sums of two or more consecutive earlier members is called the prime numbers of measurement, or sometimes the Segmented Numbers. The first few terms are 1, 2, 4, 5, 8, 10, 14, 15, 16, 21, ... (Sloane's A002048). Excluding two and three terms gives the sequence 1, 2, 4, 5, 8, 10, 12, 14, 15, 16, 19, 20, 21, ... (Sloane's A005242).
References
Guy, R. K. ``MacMahon's Prime Numbers of Measurement.'' §E30 in
Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 230-231, 1994.
Sloane, N. J. A. Sequences
A002048/M0972
and A005242/M0971
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.