This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A333379 #6 Mar 21 2020 16:35:52
%S A333379 0,1,3,6,7,14,15,26,30,31,52,58,62,63,106,116,122,126,127,212,234,244,
%T A333379 250,254,255,420,426,468,490,500,506,510,511,840,852,932,938,980,1002,
%U A333379 1012,1018,1022,1023,1700,1706,1864,1876,1956,1962,2004,2026,2036,2042
%N A333379 Numbers k such that the k-th composition in standard order is weakly increasing and covers an initial interval of positive integers.
%C A333379 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 A333379 Intersection of A333217 and A225620.
%e A333379 The sequence of terms together with the corresponding compositions begins:
%e A333379 0: () 127: (1,1,1,1,1,1,1)
%e A333379 1: (1) 212: (1,2,2,3)
%e A333379 3: (1,1) 234: (1,1,2,2,2)
%e A333379 6: (1,2) 244: (1,1,1,2,3)
%e A333379 7: (1,1,1) 250: (1,1,1,1,2,2)
%e A333379 14: (1,1,2) 254: (1,1,1,1,1,1,2)
%e A333379 15: (1,1,1,1) 255: (1,1,1,1,1,1,1,1)
%e A333379 26: (1,2,2) 420: (1,2,3,3)
%e A333379 30: (1,1,1,2) 426: (1,2,2,2,2)
%e A333379 31: (1,1,1,1,1) 468: (1,1,2,2,3)
%e A333379 52: (1,2,3) 490: (1,1,1,2,2,2)
%e A333379 58: (1,1,2,2) 500: (1,1,1,1,2,3)
%e A333379 62: (1,1,1,1,2) 506: (1,1,1,1,1,2,2)
%e A333379 63: (1,1,1,1,1,1) 510: (1,1,1,1,1,1,1,2)
%e A333379 106: (1,2,2,2) 511: (1,1,1,1,1,1,1,1,1)
%e A333379 116: (1,1,2,3) 840: (1,2,3,4)
%e A333379 122: (1,1,1,2,2) 852: (1,2,2,2,3)
%e A333379 126: (1,1,1,1,1,2) 932: (1,1,2,3,3)
%t A333379 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A333379 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333379 Select[Range[0,1000],normQ[stc[#]]&&LessEqual@@stc[#]&]
%Y A333379 Sequences covering an initial interval are counted by A000670.
%Y A333379 Compositions in standard order are A066099.
%Y A333379 Weakly increasing runs are counted by A124766.
%Y A333379 Removing the covering condition gives A225620.
%Y A333379 Removing the ordering condition gives A333217.
%Y A333379 The strictly increasing case is A164894.
%Y A333379 The strictly decreasing version is A246534.
%Y A333379 The unequal version is A333218.
%Y A333379 The weakly decreasing version is A333380.
%Y A333379 Cf. A000120, A000225, A029931, A048793, A070939, A228351, A233564, A272919.
%K A333379 nonn
%O A333379 1,3
%A A333379 _Gus Wiseman_, Mar 21 2020