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Nonarithmetic Progression Sequence

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A nonarithmetic progression sequence is an integer sequence containing no nonconstant three-term arithmetic progressions. Given two starting numbers (a_1,a_2), the sequences in the following table are obtained by repeatedly adjoining the smallest integer greater than the previous term that introduces no three-term arithmetic progression.

Sloanesequence
A0032781, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, ...
A0047931, 3, 4, 6, 10, 12, 13, 15, 28, 30, 31, 33, ...
A0331571, 4, 5, 8, 10, 13, 14, 17, 28, 31, 32, 35, ...
A0331581, 5, 6, 8, 12, 13, 17, 24, 27, 32, 34, 38, ...
A0331592, 3, 5, 6, 11, 12, 14, 15, 29, 30, 32, 33, ...
A0331602, 4, 5, 7, 11, 13, 14, 16, 29, 31, 32, 34, ...
A0331612, 5, 6, 9, 11, 14, 15, 18, 29, 32, 33, 36, ...
A0331623, 4, 6, 7, 12, 13, 15, 16, 30, 31, 33, 34, ...
A0331633, 5, 6, 8, 12, 14, 15, 17, 30, 32, 33, 35, ...
A0331644, 5, 7, 8, 13, 14, 16, 17, 31, 32, 34, 35, ...

See also

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, A004793, 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 Resource. https://mathworld.wolfram.com/NonarithmeticProgressionSequence.html

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