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A333222 Numbers k such that every restriction of the k-th composition in standard order to a subinterval has a different sum.

%I A333222 #7 Mar 18 2020 16:27:38
%S A333222 0,1,2,4,5,6,8,9,12,16,17,18,20,24,32,33,34,40,41,48,50,64,65,66,68,
%T A333222 69,70,72,80,81,88,96,98,104,128,129,130,132,133,134,144,145,160,161,
%U A333222 176,192,194,196,208,256,257,258,260,261,262,264,265,268,272,274
%N A333222 Numbers k such that every restriction of the k-th composition in standard order to a subinterval has a different sum.
%C A333222 Also numbers whose binary indices together with 0 define a Golomb ruler.
%C A333222 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.
%H A333222 Wikipedia, <a href="https://en.wikipedia.org/wiki/Golomb_ruler">Golomb ruler</a>
%e A333222 The list of terms together with the corresponding compositions begins:
%e A333222     0: ()        41: (2,3,1)    130: (6,2)      262: (6,1,2)
%e A333222     1: (1)       48: (1,5)      132: (5,3)      264: (5,4)
%e A333222     2: (2)       50: (1,3,2)    133: (5,2,1)    265: (5,3,1)
%e A333222     4: (3)       64: (7)        134: (5,1,2)    268: (5,1,3)
%e A333222     5: (2,1)     65: (6,1)      144: (3,5)      272: (4,5)
%e A333222     6: (1,2)     66: (5,2)      145: (3,4,1)    274: (4,3,2)
%e A333222     8: (4)       68: (4,3)      160: (2,6)      276: (4,2,3)
%e A333222     9: (3,1)     69: (4,2,1)    161: (2,5,1)    288: (3,6)
%e A333222    12: (1,3)     70: (4,1,2)    176: (2,1,5)    289: (3,5,1)
%e A333222    16: (5)       72: (3,4)      192: (1,7)      290: (3,4,2)
%e A333222    17: (4,1)     80: (2,5)      194: (1,5,2)    296: (3,2,4)
%e A333222    18: (3,2)     81: (2,4,1)    196: (1,4,3)    304: (3,1,5)
%e A333222    20: (2,3)     88: (2,1,4)    208: (1,2,5)    320: (2,7)
%e A333222    24: (1,4)     96: (1,6)      256: (9)        321: (2,6,1)
%e A333222    32: (6)       98: (1,4,2)    257: (8,1)      324: (2,4,3)
%e A333222    33: (5,1)    104: (1,2,4)    258: (7,2)      328: (2,3,4)
%e A333222    34: (4,2)    128: (8)        260: (6,3)      352: (2,1,6)
%e A333222    40: (2,4)    129: (7,1)      261: (6,2,1)    384: (1,8)
%t A333222 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333222 Select[Range[0,300],UnsameQ@@ReplaceList[stc[#],{___,s__,___}:>Plus[s]]&]
%Y A333222 A subset of A233564.
%Y A333222 Also a subset of A333223.
%Y A333222 These compositions are counted by A169942 and A325677.
%Y A333222 The number of distinct nonzero subsequence-sums is A333224.
%Y A333222 The number of distinct subsequence-sums is A333257.
%Y A333222 Lengths of optimal Golomb rulers are A003022.
%Y A333222 Inequivalent optimal Golomb rulers are counted by A036501.
%Y A333222 Complete rulers are A103295, with perfect case A103300.
%Y A333222 Knapsack partitions are counted by A108917, with strict case A275972.
%Y A333222 Distinct subsequences are counted by A124770 and A124771.
%Y A333222 Golomb subsets are counted by A143823.
%Y A333222 Heinz numbers of knapsack partitions are A299702.
%Y A333222 Knapsack compositions are counted by A325676.
%Y A333222 Maximal Golomb rulers are counted by A325683.
%Y A333222 Cf. A000120, A029931, A048793, A066099, A070939, A228351, A295235, A325678, A325680, A325768, A325779, A333217.
%K A333222 nonn
%O A333222 1,3
%A A333222 _Gus Wiseman_, Mar 17 2020