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A004136
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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)
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2
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1, 3, 7, 13, 21, 31, 48, 57, 73, 91, 120, 133, 168, 183
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=1..14.
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
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EXAMPLE
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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.
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CROSSREFS
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Cf. A004133, A004135.
Sequence in context: A171965 A011898 A098577 * A147409 A147342 A172310
Adjacent sequences: A004133 A004134 A004135 * A004137 A004138 A004139
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KEYWORD
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nonn,nice,more
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms and comments from Harri Haanpaa (Harri.Haanpaa(AT)hut.fi), Oct 30 2000
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STATUS
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approved
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