A004135 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 distinct elements) of which add up to a different sum (in Z_k). (Formerly M0782)

1, 2, 3, 6, 11, 19, 28, 40, 56, 72, 96, 114, 147, 178, 183, 252, 255

Offset

1,2

Comments

From Fausto A. C. Cariboni, Oct 08 2017: (Start)

Lexicographically first basis that yields a(n) for n=16:

a(16) = 252 from {0,1,2,4,32,47,54,65,94,120,128,145,169,196,217,240}

(End)

From Fausto A. C. Cariboni, Mar 12 2018: (Start)

Lexicographically first basis that yields a(n) for n=17:

a(17) = 255 from {0,1,2,4,8,16,27,32,54,64,99,108,128,141,177,198,216}

(End)

References

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..17.

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Problem C.61.

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

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

H. Haanpaa, A. Huima and Patric R. J. Östergård, 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.

H. Haanpaa, A. Huima and Patric R. J. Östergård, 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. [Annotated scanned copies of four pages only from preprint of paper]

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, (in English and Russian) [gives the term 183].

Example

a(4)=6: the set {0,1,2,4} is such a subset of Z_6, since 0+1, 0+2, 0+4, 1+2, 1+4 and 2+4 are all distinct in Z_6; also, no such 4-element set exists in any smaller cyclic group.

Crossrefs

Cf. A004136, A004133, A260998, A260999.

Sequence in context: A032156 A146385 A039827 * A288583 A090036 A024971

Adjacent sequences: A004132 A004133 A004134 * A004136 A004137 A004138

Keyword

nonn,nice,more

Author

N. J. A. Sloane

Extensions

More terms and comments from Harri Haanpaa (Harri.Haanpaa(AT)hut.fi), Nov 01 2000

a(16) from Fausto A. C. Cariboni, Oct 08 2017

a(17) from Fausto A. C. Cariboni, Mar 12 2018

Status

approved