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A004136 Additive bases: a(n) is the least integer k such that in the cyclic group Z_k there is a subset of n elements all pairs (of not necessarily distinct elements) of which add up to a different sum (in Z_k).
(Formerly M2639)
+0
3
1, 3, 7, 13, 21, 31, 48, 57, 73, 91, 120, 133, 168, 183 (list; graph; listen)
OFFSET

1,2

COMMENT

a(n) >= n^2-n+1 by a volume bound. A difference set construction by Singer shows that equality holds when n-1 is a prime power. When n is a prime power, a difference set construction by Bose shows that a(n) <= n^2-1. By computation, equality holds in the latter bound at least for 7, 11 and 13.

REFERENCES

R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404 (v_delta).

H. Haanpaa, A. Huima and P. R. J. Ostergard, Sets in Z_n with Distinct Sums of Pairs, in Optimal discrete structures and algorithms (ODSA 2000). Discrete Appl. Math. 138 (2004), no. 1-2, 99-106.

LINKS

R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs

EXAMPLE

a(3)=7: the set {0,1,3} is such a subset of Z_7, since 0+0, 0+1, 0+3, 1+1, 1+3, and 3+3 are all distinct in Z_7; also, no such 3-element set exists in any smaller cyclic group.

CROSSREFS

Cf. A004133, A004135.

Adjacent sequences: A004133 A004134 A004135 this_sequence A004137 A004138 A004139

Sequence in context: A063541 A011898 A098577 this_sequence A060939 A098575 A138035

KEYWORD

nonn,nice,more

AUTHOR

njas

EXTENSIONS

More terms and comments from Harri Haanpaa (Harri.Haanpaa(AT)hut.fi), Oct 30 2000

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Last modified October 7 14:39 EDT 2008. Contains 144666 sequences.


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