An infinite sequence of positive integers a_i satisfying


is an A-sequence if no a_k 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

 S(A)=sup_( all A sequences)sum_(k=1)^infty1/(a_k)<103.

Any A-sequence satisfies the chi inequality (Levine and O'Sullivan 1977), which gives S(A)<3.9998. Abbott (1987) and Zhang (1992) have given a bound from below, so the best result to date is


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


where chi_i are given by the Levine-O'Sullivan greedy algorithm. However, summing the first 50000 terms of the Levine-O'Sullivan sequence already gives 3.0254....

See also

B2-Sequence, Levine-O'Sullivan Greedy Algorithm, Levine-O'Sullivan Sequence, Mian-Chowla Sequence, Sum-Free Set

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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. "B_2-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 Unpublished manuscript, 1992.

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Cite this as:

Weisstein, Eric W. "A-Sequence." From MathWorld--A Wolfram Web Resource.

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