login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (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 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.

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. [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.

Sequence in context: A171965 A011898 A098577 * A147409 A147342 A172310

Adjacent sequences:  A004133 A004134 A004135 * A004137 A004138 A004139

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 12 12:50 EST 2019. Contains 329958 sequences. (Running on oeis4.)