A079467
Marks on lexicographically earliest 16-mark optimal Golomb ruler.
Original entry on oeis.org
0, 1, 4, 11, 26, 32, 56, 68, 76, 115, 117, 134, 150, 163, 168, 177
Offset: 1
A079604
Marks on lexicographically earliest 17-mark optimal Golomb ruler.
Original entry on oeis.org
0, 5, 7, 17, 52, 56, 67, 80, 81, 100, 122, 138, 159, 165, 168, 191, 199
Offset: 1
A079606
Marks on lexicographically earliest 19-mark optimal Golomb ruler.
Original entry on oeis.org
0, 1, 6, 25, 32, 72, 100, 108, 120, 130, 153, 169, 187, 190, 204, 231, 233, 242, 246
Offset: 1
A079607
Marks on lexicographically earliest 20-mark optimal Golomb ruler.
Original entry on oeis.org
0, 1, 8, 11, 68, 77, 94, 116, 121, 156, 158, 179, 194, 208, 212, 228, 240, 253, 259, 283
Offset: 1
A079608
Marks on lexicographically earliest 21-mark optimal Golomb ruler.
Original entry on oeis.org
0, 2, 24, 56, 77, 82, 83, 95, 129, 144, 179, 186, 195, 255, 265, 285, 293, 296, 310, 329, 333
Offset: 1
A079625
Marks on lexicographically earliest 22-mark optimal Golomb ruler.
Original entry on oeis.org
0, 1, 9, 14, 43, 70, 106, 122, 124, 128, 159, 179, 204, 223, 253, 263, 270, 291, 330, 341, 353, 356
Offset: 1
A079634
Marks on lexicographically earliest 23-mark optimal Golomb ruler.
Original entry on oeis.org
0, 3, 7, 17, 61, 66, 91, 99, 114, 159, 171, 199, 200, 226, 235, 246, 277, 316, 329, 348, 350, 366, 372
Offset: 1
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
A360029
Consider a ruler composed of n segments with lengths 1, 1/2, 1/3, ..., 1/n with total length A001008(n)/A002805(n). a(n) is the minimum number of distinct distances of all pairs of marks that can be achieved by permuting the positions of the segments.
Original entry on oeis.org
1, 3, 6, 10, 15, 18, 25, 33, 42, 52, 63, 71, 84, 98, 107, 123, 140, 152, 171, 185, 198, 220, 243, 256, 281, 307, 334, 354, 383, 403, 434, 466, 489, 523, 552, 581, 618, 656, 695, 728
Offset: 1
a(6) = 18: permuted segment lengths 1, 1/4, 1/2, 1/3, 1/6, 1/5 -> marks at 0, 1, 5/4, 7/4, 25/12, 9/4, 49/20 -> 18 distinct distances 1/6, 1/5, 1/4, 1/3, 11/30, 1/2, 7/10, 3/4, 5/6, 1, 13/12, 6/5, 5/4, 29/20, 7/4, 25/12, 9/4, 49/20, whereas the non-permuted ruler with marks at 0, 1, 3/2, 11/6, 25/12, 137/60, 49/20 gives 21 distinct distances 1/6, 1/5, 1/4, 1/3, 11/30, 9/20, 1/2, 7/12, 37/60, 47/60, 5/6, 19/20, 1, 13/12, 77/60, 29/20, 3/2, 11/6, 25/12, 137/60, 49/20.
-
a360029(n) = {if (n<=1, 1, my (mi=oo); w = vectorsmall(n-1, i, i+1);
forperm (w, p, my(v=vector(n,i,1/i), L=List(v)); for (m=1, n, v[m] = 1 + sum (k=1, m-1, 1/p[k]); listput(L, v[m])); for (i=1, n-1, for (j=i+1, n, listput (L, v[j]-v[i]))); mi = min(mi, #Set(L))); mi)};
A275672
Size of a largest subset of a regular cubic lattice of n*n*n points without repeated distances.
Original entry on oeis.org
0, 1, 3, 4, 6, 7, 9
Offset: 0
For n = 5, a(5) >= 7 is witnessed by {(1,1,1), (1,1,2), (1,1,4), (1,2,5), (2,3,1), (4,4,5), (5,5,4)}. There are 4223 distinct (up to rotation and reflection) 7-point configurations without repeated distances, and none of them can be extended to 8 points, so a(5) = 7.
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