Sets of integers with nonlong arithmetic progressions generated by the greedy algorithm
Authors:
Joseph L. Gerver and L. Thomas Ramsey
Journal:
Math. Comp. 33 (1979), 13531359
MSC:
Primary 10L10
MathSciNet review:
537982
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Abstract: Let be the set of positive integers containing no arithmetic progression of k terms, generated by the greedy algorithm. A heuristic formula, supported by computational evidence, is derived for the asymptotic density of in the case where k is composite. This formula, with a couple of additional assumptions, is shown to imply that the greedy algorithm would not maximize over all S with no arithmetic progression of k terms. Finally it is proved, without relying on any conjecture, that for all , the number of elements of which are less than n is greater than for sufficiently large n.
 [1]
P. ERDÖS & P. TURAN, "On certain sequences of integers," J. London Math. Soc., v. 11, 1936, pp. 261264.
 [2]
Joseph
L. Gerver, The sum of the reciprocals of a set of
integers with no arithmetic progression of 𝑘 terms, Proc. Amer. Math. Soc. 62 (1977), no. 2, 211–214. MR 0439796
(55 #12678), http://dx.doi.org/10.1090/S00029939197704397969
 [1]
 P. ERDÖS & P. TURAN, "On certain sequences of integers," J. London Math. Soc., v. 11, 1936, pp. 261264.
 [2]
 J. GERVER, "The sum of the reciprocals of a set of integers with no arithmetic progression of k terms," Proc. Amer. Math. Soc., v. 62, 1977, pp. 211214. MR 0439796 (55:12678)
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DOI:
http://dx.doi.org/10.1090/S00255718197905379820
PII:
S 00255718(1979)05379820
Article copyright:
© Copyright 1979
American Mathematical Society
