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A333255 Numbers k such that the k-th composition in standard order is strictly increasing.

%I A333255 #4 Mar 21 2020 16:35:30
%S A333255 0,1,2,4,6,8,12,16,20,24,32,40,48,52,64,72,80,96,104,128,144,160,192,
%T A333255 200,208,256,272,288,320,328,384,400,416,512,544,576,640,656,768,784,
%U A333255 800,832,840,1024,1056,1088,1152,1280,1296,1312,1536,1568,1600,1664,1680
%N A333255 Numbers k such that the k-th composition in standard order is strictly increasing.
%C A333255 A composition of n is a finite sequence of positive integers summing to n. 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 A333255 The sequence of positive terms together with the corresponding compositions begins:
%e A333255      1: (1)         128: (8)         656: (2,3,5)
%e A333255      2: (2)         144: (3,5)       768: (1,9)
%e A333255      4: (3)         160: (2,6)       784: (1,4,5)
%e A333255      6: (1,2)       192: (1,7)       800: (1,3,6)
%e A333255      8: (4)         200: (1,3,4)     832: (1,2,7)
%e A333255     12: (1,3)       208: (1,2,5)     840: (1,2,3,4)
%e A333255     16: (5)         256: (9)        1024: (11)
%e A333255     20: (2,3)       272: (4,5)      1056: (5,6)
%e A333255     24: (1,4)       288: (3,6)      1088: (4,7)
%e A333255     32: (6)         320: (2,7)      1152: (3,8)
%e A333255     40: (2,4)       328: (2,3,4)    1280: (2,9)
%e A333255     48: (1,5)       384: (1,8)      1296: (2,4,5)
%e A333255     52: (1,2,3)     400: (1,3,5)    1312: (2,3,6)
%e A333255     64: (7)         416: (1,2,6)    1536: (1,10)
%e A333255     72: (3,4)       512: (10)       1568: (1,4,6)
%e A333255     80: (2,5)       544: (4,6)      1600: (1,3,7)
%e A333255     96: (1,6)       576: (3,7)      1664: (1,2,8)
%e A333255    104: (1,2,4)     640: (2,8)      1680: (1,2,3,5)
%t A333255 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333255 Select[Range[0,1000],Less@@stc[#]&]
%Y A333255 Strictly increasing runs are counted by A124768.
%Y A333255 The normal case is A164894.
%Y A333255 The weakly decreasing version is A114994.
%Y A333255 The weakly increasing version is A225620.
%Y A333255 The unequal version is A233564.
%Y A333255 The equal version is A272919.
%Y A333255 The strictly decreasing version is A333256.
%Y A333255 Cf. A000120, A029931, A048793, A066099, A070939, A124762, A228351, A333217, A333220, A333379.
%K A333255 nonn
%O A333255 1,3
%A A333255 _Gus Wiseman_, Mar 20 2020