

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)


4



1, 3, 7, 13, 21, 31, 48, 57, 73, 91, 120, 133, 168, 183, 255, 255, 273, 307
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OFFSET

1,2


COMMENTS

a(n) >= n^2n+1 by a volume bound. A difference set construction by Singer shows that equality holds when n1 is a prime power. When n is a prime power, a difference set construction by Bose shows that a(n) <= n^21. By computation, equality holds in the latter bound at least for 7, 11, 13 and 16.
From Fausto A. C. Cariboni, Aug 13 2017: (Start)
Lexicographically first basis that yields a(n) for n=15..18:
a(15) = 255 from {0,1,3,7,15,26,31,53,63,98,107,127,140,176,197}
a(16) = 255 from {0,1,3,7,15,26,31,53,63,98,107,127,140,176,197,215}
a(17) = 273 from {0,1,3,7,15,31,63,90,116,127,136,181,194,204,233,238,255}
a(18) = 307 from {0,1,3,21,25,31,68,77,91,170,177,185,196,212,225,257,269,274}
(End)


REFERENCES

R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382404 (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. 12, 99106.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=1..18.
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See p. 162.
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
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. 12, 99106. [Annotated scanned copies of four pages only from preprint of paper]
Z. Skupien, A. Zak, Pairsums 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 131144, (in English and Russian).


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 3element set exists in any smaller cyclic group.


CROSSREFS

Cf. A004133, A004135, A260998, A260999.
Sequence in context: A171965 A011898 A098577 * A147409 A147342 A172310
Adjacent sequences: A004133 A004134 A004135 * A004137 A004138 A004139


KEYWORD

nonn,nice,more,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms and comments from Harri Haanpaa (Harri.Haanpaa(AT)hut.fi), Oct 30 2000
a(15)a(18) from Fausto A. C. Cariboni, Aug 13 2017


STATUS

approved



