Nonaveraging Sequence

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

An infinite sequence of Positive Integers

$$1\leq a_1<a_2<a_3<\ldots$$

is a nonaveraging sequence if it contains no three terms which are in an Arithmetic Progression, so that

$$a_i+a_j\not= 2a_k$$

for all distinct $a_i$, $a_j$, $a_k$. Wróblewski (1984) showed that

$$S(A)\equiv \sup_{\scriptstyle\rm all\ nonaveraging\atop\scriptstyle\rm sequences} \sum_{k=1}^\infty {1\over a_k}>3.00849.$$

References

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. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/erdos/erdos.html

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.

Wróblewski, J. ``A Nonaveraging Set of Integers with a Large Sum of Reciprocals.'' Math. Comput. 43, 261-262, 1984.