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,
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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. USA32, 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 Math19,
215-222, 1971.Wróblewski, J. "A Nonaveraging Set of Integers
with a Large Sum of Reciprocals." Math. Comput.43, 261-262, 1984.
,
,
.
The empty set and sets of length one are therefore trivially
nonaveraging.
. 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).
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