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

%I A334967 #5 Jun 04 2020 06:39:56
%S A334967 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,
%T A334967 35,36,40,42,48,56,63,64,65,66,67,68,69,70,71,72,73,74,80,81,84,85,88,
%U A334967 96,98,100,104,106,112,120,127,128,129,130,131,132,133,134
%N A334967 Numbers k such that the every subsequence (not necessarily contiguous) of the k-th composition in standard order (A066099) has a different sum.
%C A334967 First differs from A333223 in lacking 41.
%C A334967 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.
%e A334967 The sequence together with the corresponding compositions begins:
%e A334967    0: ()           18: (3,2)          48: (1,5)
%e A334967    1: (1)          19: (3,1,1)        56: (1,1,4)
%e A334967    2: (2)          20: (2,3)          63: (1,1,1,1,1,1)
%e A334967    3: (1,1)        21: (2,2,1)        64: (7)
%e A334967    4: (3)          24: (1,4)          65: (6,1)
%e A334967    5: (2,1)        26: (1,2,2)        66: (5,2)
%e A334967    6: (1,2)        28: (1,1,3)        67: (5,1,1)
%e A334967    7: (1,1,1)      31: (1,1,1,1,1)    68: (4,3)
%e A334967    8: (4)          32: (6)            69: (4,2,1)
%e A334967    9: (3,1)        33: (5,1)          70: (4,1,2)
%e A334967   10: (2,2)        34: (4,2)          71: (4,1,1,1)
%e A334967   12: (1,3)        35: (4,1,1)        72: (3,4)
%e A334967   15: (1,1,1,1)    36: (3,3)          73: (3,3,1)
%e A334967   16: (5)          40: (2,4)          74: (3,2,2)
%e A334967   17: (4,1)        42: (2,2,2)        80: (2,5)
%t A334967 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A334967 Select[Range[0,100],UnsameQ@@Total/@Union[Subsets[stc[#]]]&]
%Y A334967 These compositions are counted by A334268.
%Y A334967 Golomb rulers are counted by A169942 and ranked by A333222.
%Y A334967 Positive subset-sums of partitions are counted by A276024 and A299701.
%Y A334967 Knapsack partitions are counted by A108917 and ranked by A299702
%Y A334967 Knapsack compositions are counted by A325676 and ranked by A333223.
%Y A334967 The case of partitions is counted by A325769 and ranked by A325778.
%Y A334967 Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
%Y A334967 Number of (not necessarily contiguous) subsequences is A334299.
%Y A334967 Cf. A000120, A029931, A048793, A066099, A070939, A108917, A124771, A325770, A334300, A334967.
%K A334967 nonn
%O A334967 1,3
%A A334967 _Gus Wiseman_, Jun 02 2020