A334967 Numbers k such that the every subsequence (not necessarily contiguous) of the k-th composition in standard order (A066099) has a different sum.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 24, 26, 28, 31, 32, 33, 34, 35, 36, 40, 42, 48, 56, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 84, 85, 88, 96, 98, 100, 104, 106, 112, 120, 127, 128, 129, 130, 131, 132, 133, 134

Offset

1,3

Comments

First differs from A333223 in lacking 41.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Links

Table of n, a(n) for n=1..64.

Example

The sequence together with the corresponding compositions begins:
   0: ()           18: (3,2)          48: (1,5)
   1: (1)          19: (3,1,1)        56: (1,1,4)
   2: (2)          20: (2,3)          63: (1,1,1,1,1,1)
   3: (1,1)        21: (2,2,1)        64: (7)
   4: (3)          24: (1,4)          65: (6,1)
   5: (2,1)        26: (1,2,2)        66: (5,2)
   6: (1,2)        28: (1,1,3)        67: (5,1,1)
   7: (1,1,1)      31: (1,1,1,1,1)    68: (4,3)
   8: (4)          32: (6)            69: (4,2,1)
   9: (3,1)        33: (5,1)          70: (4,1,2)
  10: (2,2)        34: (4,2)          71: (4,1,1,1)
  12: (1,3)        35: (4,1,1)        72: (3,4)
  15: (1,1,1,1)    36: (3,3)          73: (3,3,1)
  16: (5)          40: (2,4)          74: (3,2,2)
  17: (4,1)        42: (2,2,2)        80: (2,5)

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@Total/@Union[Subsets[stc[#]]]&]

Crossrefs

These compositions are counted by A334268.

Golomb rulers are counted by A169942 and ranked by A333222.

Positive subset-sums of partitions are counted by A276024 and A299701.

Knapsack partitions are counted by A108917 and ranked by A299702

Knapsack compositions are counted by A325676 and ranked by A333223.

The case of partitions is counted by A325769 and ranked by A325778.

Contiguous subsequence-sums are counted by A333224 and ranked by A333257.

Number of (not necessarily contiguous) subsequences is A334299.

Cf. A000120, A029931, A048793, A066099, A070939, A108917, A124771, A325770, A334300, A334967.

Sequence in context: A120003 A353852 A333223 * A036965 A133184 A102576

Adjacent sequences: A334964 A334965 A334966 * A334968 A334969 A334970

Keyword

nonn

Author

Gus Wiseman, Jun 02 2020

Status

approved