A260999 -id:A260999 - cp's OEIS frontend

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).

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

Offset: 1

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)

			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.

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

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].

Cf. A004136, A004133, A260998, A260999.

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

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).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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, 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)
Such sets are also known as modular Golomb rulers, or circular Golomb rulers. - Andrey Zabolotskiy, Sep 11 2017

			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.

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

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, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404 (v_delta).
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 above]
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).

Cf. A004133, A004135, A260998, A260999, A003022.

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

A260998 Maximal size of a subset of Z_n with distinct sums of pairs (of distinct elements).

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

N. J. A. Sloane, Aug 10 2015

nonn

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..254
Fausto A. C. Cariboni, S_2-sets that yield a(n) for n = 2..254, Mar 24 2018.
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]
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.

Cf. A004135, A004136, A260999.

By the pigeonhole principle, C(a(n),2) <= n, yielding upper bound a(n) <= floor((1+sqrt(8*n+1))/2). - Rob Pratt, Nov 27 2017

a(1)-a(90) from H. Haanpaa, A. Huima and Patric R. J. Östergård (see link), Nov 08 2000

a(1)-a(90) confirmed by Fausto A. C. Cariboni, Nov 09 2017

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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).

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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).

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A260998 Maximal size of a subset of Z_n with distinct sums of pairs (of distinct elements).

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