Nonaveraging Sequence

A sequence of positive integers

(1)

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

(2)

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

Consider all possible subsets on the integers . There is one nonaveraging sequence on  (), two on  ( and ), four on , and so on. For example, 13 of the 16 subjects of  are nonaveraging, with , , and  excluded. The numbers of nonaveraging subsets on , , ... are 1, 2, 4, 7, 13, 23, 40, ... (OEIS A051013).

Wróblewski (1984) showed that for infinite nonaveraging sequences,

(3)

REFERENCES:

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  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.