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Search: a260998 -id:a260998
Displaying 1-3 of 3 results found.
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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).
(Formerly M0782)
+10
7
1, 2, 3, 6, 11, 19, 28, 40, 56, 72, 96, 114, 147, 178, 183, 252, 255
(list; graph; refs; listen; history; text; internal format)
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.
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
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)
+10
5
1, 3, 7, 13, 21, 31, 48, 57, 73, 91, 120, 133, 168, 183, 255, 255, 273, 307
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
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..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, 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).
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 3-element set exists in any smaller cyclic group.
CROSSREFS
Cf. A004133, A004135, A260998, A260999, A003022.
KEYWORD
nonn,nice,more
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
A260999
Maximal size of a subset of Z_n with distinct sums of any two elements.
+10
4
1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 8, 8, 8, 8, 8, 8, 9, 8, 8, 8, 8
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 1..295
Fausto A. C. Cariboni, Sets that yield a(n) for n = 2..295, Jan 27 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.
FORMULA
By the pigeonhole principle, C(a(n)+1,2) <= n, yielding upper bound a(n) <= floor((-1+sqrt(8*n+1))/2). - Rob Pratt, Nov 27 2017
CROSSREFS
Cf. A004135, A004136, A260998.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 10 2015
EXTENSIONS
More terms from Rob Pratt, Nov 27 2017
STATUS
approved
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