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B_2-Sequence

A B_2-sequence is an infinite sequence of positive integers

1<=b_1<b_2<b_3<...,
(1)

also called a Sidon sequence, such that all pairwise sums

b_i+b_j
(2)

for i<=j are distinct (Guy 1994). An example is 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, ... (OEIS A005282). Halberstam and Roth (1983) contains an accessible account of most known results up to around 1965. Recent advances have been made by Cilleruelo, Jia, Kolountzakis, Lindstrom, and Ruzsa.

Zhang (1993, 1994) showed that

S(B_2)=sup_(all B_2 sequences)sum_(k=1)^infty1/(b_k)>2.1597,
(3)

which has been increased to S(B_2)>2.16086 by R. Lewis using the B_2-sequence 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 291, 324, ... (OEIS A046185). L. J. Kleinwaks (pers. comm., June 16, 2025) improved the lower bound to

S(B_2)>2.1615
(4)

using the sequence 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, ... (OEIS A384729). The definition can be extended to B_n-sequences (Guy 1994).

See also

A-Sequence, Mian-Chowla Sequence, Sidon Set

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References

Finch, S. R. "Erdős' Reciprocal Sum Constants." §2.20 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 163-166, 2003.Guy, R. K. "Packing Sums of Pairs," "Three-Subsets with Distinct Sums," "B_2-Sequences," and "B_2-Sequences Formed by the Greedy Algorithm." §C9, C11, E28, and E32 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 115-118, 121-123, 228-229, and 232-233, 1994.Halberstam, H. and Roth, K. Sequences, rev. ed. New York: Springer-Verlag, 1983.Mian, A. M. and Chowla, S. D. "On the B_2-Sequences of Sidon." Proc. Nat. Acad. Sci. India A14, 3-4, 1944.O'Bryant, K. "A Complete Annotated Bibliography of Work Related to Sidon Sequences." Electron. J. Combin. Dynamic Survey 11, 2004. https://doi.org/10.37236/32.Sloane, N. J. A. Sequences A005282/M1094, A046185, and A384729 in "The On-Line Encyclopedia of Integer Sequences."Zhang, Z. X. "A B2-Sequence with Larger Reciprocal Sum." Math. Comput. 60, 835-839, 1993.Zhang, Z. X. "Finding Finite B2-Sequences with Larger m-a_m^(1/2)." Math. Comput. 63, 403-414, 1994.

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B_2-Sequence

Cite this as:

Weisstein, Eric W. "B_2-Sequence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/B2-Sequence.html

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