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).
Original entry on oeis.org
1, 2, 3, 6, 11, 19, 28, 40, 56, 72, 96, 114, 147, 178, 183, 252, 255
Offset: 1
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].
More terms and comments from Harri Haanpaa (Harri.Haanpaa(AT)hut.fi), Nov 01 2000
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
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).
More terms and comments from Harri Haanpaa (Harri.Haanpaa(AT)hut.fi), Oct 30 2000
A232234
Additive bases: a(n) is the least integer such that there is an n-element set of nonnegative integers, the sums of pairs of which are distinct and at most a(n).
Original entry on oeis.org
0, 2, 6, 12, 22, 34, 50, 68, 88, 110, 144, 170, 212, 254, 302, 354, 398, 432, 492, 566, 666, 712, 744, 850, 960, 984, 1106
Offset: 1
These terms are twice the terms of
A003022.
A345731
Additive bases: a(n) is the least integer such that there is an n-element set of integers between 0 and a(n), the sums of pairs (of distinct elements) of which are distinct.
Original entry on oeis.org
1, 2, 4, 7, 12, 18, 24, 34, 45, 57, 71, 86, 105, 126, 148
Offset: 2
a(6)=12 because 0-1-2-4-7-12 (0-5-8-10-11-12) resp. 0-1-2-6-9-12 (0-3-6-10-11-12) are shortest weak Sidon sets of size 6.
a(16)=148: [0, 3, 5, 6, 32, 49, 59, 68, 93, 106, 118, 126, 130, 134, 141, 148]. - _Zhao Hui Du_, Jul 27 2025
- Alison M. Marr and W. D. Wallis, Magic Graphs, Birkhäuser, 2nd ed., 2013. See Section 2.3.
- Xiaodong Xu, Meilian Liang, and Zehui Shao, On weak Sidon sequences, The Journal of Combinatorial Mathematics and Combinatorial Computing (2014), 107--113
-
a[n_Integer?NonNegative] := Module[{k = n - 1}, While[SelectFirst[Subsets[Range[0, k - 1], {n - 1}], Length@Union[Plus @@@ Subsets[#~Join~{k}, {2}]] >= (n*(n - 1))/2 &] === Missing["NotFound"], k++]; k];
Table[a[n], {n, 2, 8}] (* Robert P. P. McKone, Nov 05 2023 *)
-
from itertools import combinations, count
def a(n):
for k in count(n-1):
for c in combinations(range(k), n-1):
c = c + (k,)
ss = set()
for s in combinations(c, 2):
if sum(s) in ss: break
else: ss.add(sum(s))
if len(ss) == n*(n-1)//2: return k # use (k, c) for sets
print([a(n) for n in range(2, 9)]) # Michael S. Branicky, Jun 25 2021
a(16) corrected and a(17) deleted by
Zhao Hui Du, Jul 27 2025
Showing 1-4 of 4 results.
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