A003022 -id:A003022 - cp's OEIS frontend

A382382 Least k for which there exists an n-subset X of {0, ..., k} such that the variances of the subsets of X of size at least 2 are distinct.

0, 1, 3, 6, 11, 17, 27, 48

Offset: 1

Pontus von Brömssen, Mar 23 2025

The variance of a nonempty set Y is (Sum_{y in Y} (y-m)^2)/|Y|, where m is the average of Y and |Y| is the size of Y.

0 and a(n) necessarily belong to the set X in the definition.

			    | a set X that satisfy the condition
  n | (the largest element of X is a(n))
  --+-----------------------------------
  1 | {0}
  2 | {0, 1}
  3 | {0, 1, 3}
  4 | {0, 1, 4,  6}
  5 | {0, 2, 7,  8, 11}
  6 | {0, 1, 4, 10, 12, 17}
  7 | {0, 3, 4, 14, 19, 21, 27}
  8 | {0, 1, 5, 15, 22, 40, 46, 48}

Cf. A003022, A382381, A382383.

A003022(n) <= a(n) < A382381(n) for n >= 2.

A079434 Marks on lexicographically earliest 13-mark optimal Golomb ruler.

Original entry on oeis.org

0, 2, 5, 25, 37, 43, 59, 70, 85, 89, 98, 99, 106

Offset: 1

David W. Wilson, Feb 16 2003

J. B. Shearer, Golomb ruler table
Eric Weisstein's World of Mathematics, Golomb rulers.
Index entries for sequences related to Golomb rulers

Cf. A003022.

A079605 Marks on lexicographically earliest 18-mark optimal Golomb ruler.

Original entry on oeis.org

0, 2, 10, 22, 53, 56, 82, 83, 89, 98, 130, 148, 153, 167, 188, 192, 205, 216

Offset: 1

David W. Wilson, Feb 16 2003

J. B. Shearer, Golomb ruler table
Eric Weisstein's World of Mathematics, Golomb rulers.
Index entries for sequences related to Golomb rulers

Cf. A003022.

A227358 Length of shortest Golomb-like (for sums of triples) ruler with n marks.

Original entry on oeis.org

0, 1, 4, 11, 23, 45, 82, 129, 208, 309

Offset: 1

John Tromp, Jul 08 2013

a(n) is the least integer such that there is an n-element set of integers between 0 and a(n), the sums of triples (of not necessarily distinct elements) of which are distinct.

a(11) = 445 or a(11) < 440, but disproving the latter will take many cpu-years with the given program. - John Tromp, Aug 28 2013

			a(4) = 11 because 0-1-7-11 (0-4-10-11) and 0-1-8-11 (0-3-10-11) have all (6 choose 3)=20 distinct triple sums and there is no 0=b0<b1<b2<b3<11 with distinct triple sums.

IBM Research, Ponder This, July 2013 essentially asks for a(8).
Kevin O'Bryant, Constructing Thick B_h sets, arXiv:2308.12406 [math.NT], 2023.
John Tromp, PonderThisJuly2013.c

Cf. A003022, A227588.

C
```
// See link.
```

a(n) = A227588(n,3) - 1. - James Wilcox, Aug 02 2013

a(8)-a(10) from John Tromp, Jul 30 2013

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

N. J. A. Sloane, Nov 24 2013

By definition, these terms are twice the terms of A003022, see comment there. - Bernd Mulansky, Jun 25 2021
Lexicographically first basis that yields a(16) = 354 is {0,1,4,11,26,32,56,68,76,115,117,134,150,163,168,177}. - Fausto A. C. Cariboni, Nov 01 2017
Lexicographically first basis that yields a(17) = 398 is {0,5,7,17,52,56,67,80,81,100,122,138,159,165,168,191,199}. - Fausto A. C. Cariboni, Nov 26 2017

Fausto A. C. Cariboni, All additive bases that yield a(n) for n = 2..17, Dec 31 2017.
Z. Skupien and 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).

These terms are twice the terms of A003022.

See A004133 for another version.

a(16) from Fausto A. C. Cariboni, Nov 01 2017
a(17) from Fausto A. C. Cariboni, Nov 26 2017
a(18)-a(27) added by Bernd Mulansky, Jun 25 2021

A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 5, 10, 10, 24, 24, 43, 42, 88, 72, 136, 122, 242, 213, 392, 320, 630, 490, 916, 742, 1432, 1160, 1955, 1604, 2826, 2310, 3850, 2888, 5416, 4426, 7332, 5814, 10046, 7983, 12946, 10236, 17780, 14100, 22674, 17582, 30232, 23674, 37522, 29426, 49832

Offset: 0

Gus Wiseman, Jun 02 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

The contiguous case is A325676.

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,2)    (1,3)      (1,4)        (1,5)
              (2,1)    (2,2)      (2,3)        (2,4)
              (1,1,1)  (3,1)      (3,2)        (3,3)
                       (1,1,1,1)  (4,1)        (4,2)
                                  (1,1,3)      (5,1)
                                  (1,2,2)      (1,1,4)
                                  (2,2,1)      (2,2,2)
                                  (3,1,1)      (4,1,1)
                                  (1,1,1,1,1)  (1,1,1,1,1,1)

These compositions are ranked by A334967.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and A325592 and ranked by A299702, while the strict case is counted by A275972 and ranked by A059519 and A301899.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A003022, A029931, A103295, A143823, A325680, A333224.

b:= proc(n, s) option remember; `if`(n=0, 1, add((h->
      `if`(nops(h)=nops(map(l-> add(i, i=l), h)),
       b(n-j, h), 0))({s[], map(l-> [l[], j], s)[]}), j=1..n))
    end:
a:= n-> b(n, {[]}):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 03 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&]],{n,0,15}]

a(18)-a(47) from Alois P. Heinz, Jun 03 2020

A335279 Positions of first appearances in A124771 = number of distinct contiguous subsequences of compositions in standard order.

Original entry on oeis.org

0, 1, 3, 5, 11, 15, 23, 27, 37, 47, 55, 107, 111, 119, 155, 215, 223, 239, 411, 431, 471, 479, 495, 549, 631, 943, 951, 959, 991, 1647, 1887, 1967, 1983, 2015, 2543, 2935, 3703, 3807, 3935, 3967, 4031, 6639, 6895, 7407, 7871, 7903, 8063, 8127, 10207, 13279

Offset: 1

Gus Wiseman, Jun 03 2020

nonn

			The sequence together with the corresponding compositions begins:
     0: ()                215: (1,2,2,1,1,1)
     1: (1)               223: (1,2,1,1,1,1,1)
     3: (1,1)             239: (1,1,2,1,1,1,1)
     5: (2,1)             411: (1,3,1,2,1,1)
    11: (2,1,1)           431: (1,2,2,1,1,1,1)
    15: (1,1,1,1)         471: (1,1,2,2,1,1,1)
    23: (2,1,1,1)         479: (1,1,2,1,1,1,1,1)
    27: (1,2,1,1)         495: (1,1,1,2,1,1,1,1)
    37: (3,2,1)           549: (4,3,2,1)
    47: (2,1,1,1,1)       631: (3,1,1,2,1,1,1)
    55: (1,2,1,1,1)       943: (1,1,2,2,1,1,1,1)
   107: (1,2,2,1,1)       951: (1,1,2,1,2,1,1,1)
   111: (1,2,1,1,1,1)     959: (1,1,2,1,1,1,1,1,1)
   119: (1,1,2,1,1,1)     991: (1,1,1,2,1,1,1,1,1)
   155: (3,1,2,1,1)      1647: (1,3,1,2,1,1,1,1)
The subsequences for n = 0, 1, 3, 5, 11, 15, 23, 27 are the following (0 = empty partition):
  0  0  0   0   0    0     0     0     0    0
     1  1   1   1    1     1     1     1    1
        11  2   2    11    2     2     2    2
            21  11   111   11    11    3    11
                21   1111  21    12    21   21
                211        111   21    32   111
                           211   121   321  211
                           2111  211        1111
                                 1211       2111
                                            21111

Positions of first appearances in A124771.
Compositions where every subinterval has a different sum are A333222.
Knapsack compositions are A333223.
Cf. A000120, A003022, A029931, A066099, A070939, A124767, A124770, A325770, A334299, A334968.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
seq=Table[Length[Union[ReplaceList[stc[n],{_,s___,_}:>{s}]]],{n,0,1000}];
Table[Position[seq,i][[1,1]]-1,{i,First/@Gather[seq]}]

A031873 Lexically first 10-mark Golomb ruler.

Original entry on oeis.org

1, 2, 7, 11, 24, 27, 35, 42, 54, 56

Offset: 1

Daniel Smith (dsmith(AT)globalnet.co.uk)

An n-mark Golomb ruler is a set of integers a(1) < a(2) < ... < a(n) such that [1] i < j, a(j) - a(i) = a(n) - a(m) => (i, j) = (m, n) and [2] a(n) - a(1) is the smallest possible value permitting [1].

Index entries for sequences related to Golomb rulers

Cf. A003022. Equals A079426 + 1.

Better description supplied by David W. Wilson, Feb 16 2003

A079435 Marks on lexicographically earliest 14-mark optimal Golomb ruler.

Original entry on oeis.org

0, 4, 6, 20, 35, 52, 59, 77, 78, 86, 89, 99, 122, 127

Offset: 1

David W. Wilson, Feb 16 2003

J. B. Shearer, Golomb ruler table
Eric Weisstein's World of Mathematics, Golomb rulers.
Index entries for sequences related to Golomb rulers

Cf. A003022.

A079454 Marks on lexicographically earliest 15-mark optimal Golomb ruler.

Original entry on oeis.org

0, 4, 20, 30, 57, 59, 62, 76, 100, 111, 123, 136, 144, 145, 151

Offset: 1

David W. Wilson, Feb 16 2003

J. B. Shearer, Golomb ruler table
Eric Weisstein's World of Mathematics, Golomb rulers.
Index entries for sequences related to Golomb rulers

Cf. A003022.

cp's OEIS Frontend

A382382 Least k for which there exists an n-subset X of {0, ..., k} such that the variances of the subsets of X of size at least 2 are distinct.

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A079434 Marks on lexicographically earliest 13-mark optimal Golomb ruler.

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A079605 Marks on lexicographically earliest 18-mark optimal Golomb ruler.

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A227358 Length of shortest Golomb-like (for sums of triples) ruler with n marks.

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C

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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).

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A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

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Maple

Mathematica

Extensions

A335279 Positions of first appearances in A124771 = number of distinct contiguous subsequences of compositions in standard order.

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Examples

Crossrefs

Programs

Mathematica

A031873 Lexically first 10-mark Golomb ruler.

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Keywords

Comments

Links

Crossrefs

Extensions

A079435 Marks on lexicographically earliest 14-mark optimal Golomb ruler.

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Keywords

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Crossrefs

A079454 Marks on lexicographically earliest 15-mark optimal Golomb ruler.

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Keywords

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