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A333380 Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.

%I A333380 #4 Mar 21 2020 16:36:01
%S A333380 0,1,3,5,7,11,15,21,23,31,37,43,47,63,75,85,87,95,127,149,151,171,175,
%T A333380 191,255,293,299,303,341,343,351,383,511,549,587,597,599,607,683,687,
%U A333380 703,767,1023,1099,1173,1175,1195,1199,1215,1365,1367,1375,1407,1535
%N A333380 Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.
%C A333380 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.
%F A333380 Intersection of A333217 and A114994.
%e A333380 The sequence of terms together with the corresponding compositions begins:
%e A333380     0: ()               127: (1,1,1,1,1,1,1)
%e A333380     1: (1)              149: (3,2,2,1)
%e A333380     3: (1,1)            151: (3,2,1,1,1)
%e A333380     5: (2,1)            171: (2,2,2,1,1)
%e A333380     7: (1,1,1)          175: (2,2,1,1,1,1)
%e A333380    11: (2,1,1)          191: (2,1,1,1,1,1,1)
%e A333380    15: (1,1,1,1)        255: (1,1,1,1,1,1,1,1)
%e A333380    21: (2,2,1)          293: (3,3,2,1)
%e A333380    23: (2,1,1,1)        299: (3,2,2,1,1)
%e A333380    31: (1,1,1,1,1)      303: (3,2,1,1,1,1)
%e A333380    37: (3,2,1)          341: (2,2,2,2,1)
%e A333380    43: (2,2,1,1)        343: (2,2,2,1,1,1)
%e A333380    47: (2,1,1,1,1)      351: (2,2,1,1,1,1,1)
%e A333380    63: (1,1,1,1,1,1)    383: (2,1,1,1,1,1,1,1)
%e A333380    75: (3,2,1,1)        511: (1,1,1,1,1,1,1,1,1)
%e A333380    85: (2,2,2,1)        549: (4,3,2,1)
%e A333380    87: (2,2,1,1,1)      587: (3,3,2,1,1)
%e A333380    95: (2,1,1,1,1,1)    597: (3,2,2,2,1)
%t A333380 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A333380 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333380 Select[Range[0,1000],normQ[stc[#]]&&GreaterEqual@@stc[#]&]
%Y A333380 Sequences covering an initial interval are counted by A000670.
%Y A333380 Compositions in standard order are A066099.
%Y A333380 Weakly decreasing runs are counted by A124765.
%Y A333380 Removing the covering condition gives A114994.
%Y A333380 Removing the ordering condition gives A333217.
%Y A333380 The strictly decreasing case is A246534.
%Y A333380 The unequal version is A333218.
%Y A333380 The weakly increasing version is A333379.
%Y A333380 Cf. A000120, A000225, A029931, A048793, A070939, A164894, A225620, A228351, A233564, A272919, A333219.
%K A333380 nonn
%O A333380 1,3
%A A333380 _Gus Wiseman_, Mar 21 2020