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%I #7 Mar 18 2020 23:02:14
%S 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 35,36,40,41,42,48,50,56,63,64,65,66,67,68,69,70,71,72,73,74,80,81,84,
%U 85,88,96,98,100,104,106,112,120,127,128,129,130,131,132,133
%N Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.
%C The k-th composition in standard order (row k of 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.
%e The list of terms together with the corresponding compositions begins:
%e 0: () 21: (2,2,1) 65: (6,1)
%e 1: (1) 24: (1,4) 66: (5,2)
%e 2: (2) 26: (1,2,2) 67: (5,1,1)
%e 3: (1,1) 28: (1,1,3) 68: (4,3)
%e 4: (3) 31: (1,1,1,1,1) 69: (4,2,1)
%e 5: (2,1) 32: (6) 70: (4,1,2)
%e 6: (1,2) 33: (5,1) 71: (4,1,1,1)
%e 7: (1,1,1) 34: (4,2) 72: (3,4)
%e 8: (4) 35: (4,1,1) 73: (3,3,1)
%e 9: (3,1) 36: (3,3) 74: (3,2,2)
%e 10: (2,2) 40: (2,4) 80: (2,5)
%e 12: (1,3) 41: (2,3,1) 81: (2,4,1)
%e 15: (1,1,1,1) 42: (2,2,2) 84: (2,2,3)
%e 16: (5) 48: (1,5) 85: (2,2,2,1)
%e 17: (4,1) 50: (1,3,2) 88: (2,1,4)
%e 18: (3,2) 56: (1,1,4) 96: (1,6)
%e 19: (3,1,1) 63: (1,1,1,1,1,1) 98: (1,4,2)
%e 20: (2,3) 64: (7) 100: (1,3,3)
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Select[Range[0,100],UnsameQ@@Total/@Union[ReplaceList[stc[#],{___,s__,___}:>{s}]]&]
%Y Distinct subsequences are counted by A124770 and A124771.
%Y A superset of A333222, counted by A169942, with partition case A325768.
%Y These compositions are counted by A325676.
%Y A version for partitions is A325769, with Heinz numbers A325778.
%Y The number of distinct positive subsequence-sums is A333224.
%Y The number of distinct subsequence-sums is A333257.
%Y Numbers whose binary indices are a strict knapsack partition are A059519.
%Y Knapsack partitions are counted by A108917, with strict case A275972.
%Y Golomb subsets are counted by A143823.
%Y Heinz numbers of knapsack partitions are A299702.
%Y Maximal Golomb rulers are counted by A325683.
%Y Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295 A325779, A233564, A325680, A325687, A325770, A333217.
%K nonn
%O 1,3
%A _Gus Wiseman_, Mar 17 2020
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