Nonaveraging Sequence

A sequence of positive integers


is a nonaveraging sequence if it contains no three terms which are in an arithmetic progression, i.e., terms such that


for distinct a_i, a_j, a_k. The empty set and sets of length one are therefore trivially nonaveraging.

Consider all possible subsets on the integers S_n={1,2,...,n}. There is one nonaveraging sequence on S_0 (emptyset), two on S_1 (emptyset and {1}), four on S_2, and so on. For example, 13 of the 16 subjects of S_4 are nonaveraging, with {1,2,3}, {2,3,4}, and {1,2,3,4} excluded. The numbers of nonaveraging subsets on S_0, S_1, ... are 1, 2, 4, 7, 13, 23, 40, ... (OEIS A051013).

Wróblewski (1984) showed that for infinite nonaveraging sequences,

 S(A)=sup_(all nonaveraging; sequences)sum_(k=1)^infty1/(a_k)>3.00849.

See also

A-Sequence, Nondividing Set

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Abbott, H. L. "On a Conjecture of Erdős and Straus on Non-Averaging Sets of Integers." In Proceedings of the Fifth British Combinatorial Conference, University of Aberdeen, Aberdeen, July 14-18, 1975 (Ed. C. St. J. A. Nash-Williams and J. Sheehan). Winnipeg, Manitoba, Canada: Utilitas Math. Pub., pp. 1-4, 1976.Abbott, H. L. "Extremal Problems on Non-Averaging and Non-Dividing Sets." Pacific J. Math. 91, 1-12, 1980.Abbott, H. L. "On the Erdős-Straus Non-Averaging Set Problem." Acta Math. Hungar. 47, 117-119, 1986.Behrend, F. "On Sets of Integers which Contain no Three Terms in an Arithmetic Progression." Proc. Nat. Acad. Sci. USA 32, 331-332, 1946.Finch, S. R. "Erdős' Reciprocal Sum Constants." §2.20 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 163-166, 2003.Gerver, J. L. "The Sum of the Reciprocals of a Set of Integers with No Arithmetic Progression of k Terms." Proc. Amer. Math. Soc. 62, 211-214, 1977.Gerver, J. L. and Ramsey, L. "Sets of Integers with no Long Arithmetic Progressions Generated by the Greedy Algorithm." Math. Comput. 33, 1353-1360, 1979.Guy, R. K. "Nonaveraging Sets. Nondividing Sets." §C16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 131-132, 1994.Sloane, N. J. A. Sequence A051013 in "The On-Line Encyclopedia of Integer Sequences."Straus, E. G. "Non-Averaging Sets." Proc. Symp. Pure Math 19, 215-222, 1971.Wróblewski, J. "A Nonaveraging Set of Integers with a Large Sum of Reciprocals." Math. Comput. 43, 261-262, 1984.

Referenced on Wolfram|Alpha

Nonaveraging Sequence

Cite this as:

Weisstein, Eric W. "Nonaveraging Sequence." From MathWorld--A Wolfram Web Resource.

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