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

Author

Gus Wiseman, Jun 02 2020

Keywords

• nonn

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.

Examples

			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)
		

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.

Programs

Mathematica

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