also called a Sidon sequence, such that all pairwise sums
(2)
for 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
(3)
which has been increased to by R. Lewis using the non- sequence 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148,
182, 204, 252, 291, 324, ... (OEIS A046185).
The definition can be extended to -sequences (Guy 1994).
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," "-Sequences," and "-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 -Sequences of Sidon." Proc. Nat. Acad. Sci. IndiaA14,
3-4, 1944.Sloane, N. J. A. Sequences A005282/M1094
and A046185 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 ." Math. Comput.63, 403-414,
1994.