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# 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, ...

Arithmetic Progression

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## References

Allouche, J.-P. and Shallit, J. "The Ring of k-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 in Unsolved 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. Sequences A003278/M0975, A033156, A033157, A033158, A033159, A033160, A033161, A033162, A033163, and A033164 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Nonarithmetic Progression Sequence

## Cite this as:

Weisstein, Eric W. "Nonarithmetic Progression Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NonarithmeticProgressionSequence.html