TOPICS
Nonarithmetic Progression Sequence
Given two starting numbers
, the following table gives the unique sequences
that contain no three-term arithmetic progressions.
| Sloane | sequence |
| A003278 | 1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, ... |
| A033156 | 1, 3, 4, 6, 10, 12, 13, 15, 28, 30, 31, 33, ... |
| A033157 | 1, 4, 5, 8, 10, 13, 14, 17, 28, 31, 32, 35, ... |
| A033158 | 1, 5, 6, 8, 12, 13, 17, 24, 27, 32, 34, 38, ... |
| A033159 | 2, 3, 5, 6, 11, 12, 14, 15, 29, 30, 32, 33, ... |
| A033160 | 2, 4, 5, 7, 11, 13, 14, 16, 29, 31, 32, 34, ... |
| A033161 | 2, 5, 6, 9, 11, 14, 15, 18, 29, 32, 33, 36, ... |
| A033162 | 3, 4, 6, 7, 12, 13, 15, 16, 30, 31, 33, 34, ... |
| A033163 | 3, 5, 6, 8, 12, 14, 15, 17, 30, 32, 33, 35, ... |
| A033164 | 4, 5, 7, 8, 13, 14, 16, 17, 31, 32, 34, 35, ... |
See also
Arithmetic ProgressionExplore with Wolfram|Alpha
More things to try:
References
Allouche, J.-P. and Shallit, J. "The Ring ofk-Regular Sequences."Theor. Comput. Sci.98, 163-197, 1992.Erdős, P. and Turán, P. "On Some Sequences of Integers."J. London Math. Soc.11, 261-264, 1936.Gerver, J.; Propp, J.; and Simpson, J. "Greedily Partitioning the Natural Numbers into Sets Free of Arithmetic Progressions."Proc. Amer. Math. Soc.102, 765-772, 1988.Guy, R. K. "Theorem of van der Waerden, Szemerédi's Theorem. Partitioning the Integers into Classes; at Least One Contains an A.P." §E10 inUnsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 204-209, 1994.Iacobescu, F. "Smarandache Partition Type and Other Sequences."Bull. Pure Appl. Sci.16E, 237-240, 1997.Ibstedt, H. "A Few Smarandache Sequences."Smarandache Notions J.8, 170-183, 1997.Sloane, N. J. A. SequencesA003278/M0975,A033156,A033157,A033158,A033159,A033160,A033161,A033162,A033163, andA033164 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Nonarithmetic Progression SequenceCite this as:
Weisstein, Eric W. "Nonarithmetic ProgressionSequence." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/NonarithmeticProgressionSequence.html