|
Search: id:A004136
|
|
|
| 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)
|
|
+0
3
|
|
| 1, 3, 7, 13, 21, 31, 48, 57, 73, 91, 120, 133, 168, 183
(list; graph; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
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 and 13.
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
|
|
LINKS
|
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
|
|
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.
Sequence in context: A063541 A011898 A098577 this_sequence A147409 A147342 A060939
Adjacent sequences: A004133 A004134 A004135 this_sequence A004137 A004138 A004139
|
|
KEYWORD
|
nonn,nice,more
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
EXTENSIONS
|
More terms and comments from Harri Haanpaa (Harri.Haanpaa(AT)hut.fi), Oct 30 2000
|
|
|
Search completed in 0.002 seconds
|
