A-Sequence

An infinite sequence of positive integers  satisfying

(1)

is an -sequence if no  is the sum of two or more distinct earlier terms (Guy 1994). Such sequences are sometimes also known as sum-free sets.

Erdős (1962) proved

(2)

Any -sequence satisfies the chi inequality (Levine and O'Sullivan 1977), which gives . Abbott (1987) and Zhang (1992) have given a bound from below, so the best result to date is

(3)

Levine and O'Sullivan (1977) conjectured that the sum of reciprocals of an -sequence satisfies

(4)

where  are given by the Levine-O'Sullivan greedy algorithm. However, summing the first  terms of the Levine-O'Sullivan sequence already gives 3.0254....

REFERENCES:

Abbott, H. L. "On Sum-Free Sequences." Acta Arith. 48, 93-96, 1987.

Erdős, P. "Remarks on Number Theory III. Some Problems in Additive Number Theory." Mat. Lapok 13, 28-38, 1962.

Finch, S. R. "Erdős' Reciprocal Sum Constants." §2.20 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 163-166, 2003.

Guy, R. K. "-Sequences." §E28 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 228-229, 1994.

Levine, E. and O'Sullivan, J. "An Upper Estimate for the Reciprocal Sum of a Sum-Free Sequence." Acta Arith. 34, 9-24, 1977.

Zhang, Z. X. "A Sum-Free Sequence with Larger Reciprocal Sum." Unpublished manuscript, 1992.