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

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. Lapok13,
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 Unpublished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.