A-Sequence

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

An Infinite Sequence of Positive Integers $a_i$ satisfying

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

is an $A$-sequence if no $a_k$ is the Sum of two or more distinct earlier terms (Guy 1994). Erdös (1962) proved

$$S(A)\equiv \sup_{\rm\ all\ {\it A}\ sequences}\, \sum_{k=1}^\infty {1\over a_k}<103.$$ (2)

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

$$2.0649<S(A)<3.9998.$$ (3)

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

$$S(A)\leq \sum_{k=1}^\infty {1\over \chi_k}=3.01\ldots,$$ (4)

where $\chi_i$ are given by the Levine-O'Sullivan Greedy Algorithm.

See also B2-Sequence, Mian-Chowla Sequence

References

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

Erdös, P. ``Remarks on Number Theory III. Some Problems in Additive Number Theory.'' Mat. Lapok 13, 28-38, 1962.

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

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 Unpublished manuscript, 1992.