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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)
1, 3, 7, 13, 21, 31, 48, 57, 73, 91, 120, 133, 168, 183 (list; graph; refs; listen; history; text; internal format)



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.


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.

Z. Skupien, A. Zak, Pair-sums packing and rainbow cliques, in TOPICS IN GRAPH THEORY, A tribute to A. A. and T. E. Zykovs on the occasion of A. A. Zykov's 90th birthday, ed. R. Tyshkevich, Univ. Illinois, 2013, pages 131-144; http://www.math.uiuc.edu/~kostochk/Zykov90-Topics_in_Graph_Theory.pdf

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


Table of n, a(n) for n=1..14.

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


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.


Cf. A004133, A004135.

Sequence in context: A171965 A011898 A098577 * A147409 A147342 A172310

Adjacent sequences:  A004133 A004134 A004135 * A004137 A004138 A004139




N. J. A. Sloane.


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



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Last modified August 1 12:10 EDT 2015. Contains 260180 sequences.