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

0, 1, 3, 5, 7, 11, 15, 21, 23, 31, 37, 43, 47, 63, 75, 85, 87, 95, 127, 149, 151, 171, 175, 191, 255, 293, 299, 303, 341, 343, 351, 383, 511, 549, 587, 597, 599, 607, 683, 687, 703, 767, 1023, 1099, 1173, 1175, 1195, 1199, 1215, 1365, 1367, 1375, 1407, 1535

Offset: 1

Author

Gus Wiseman, Mar 21 2020

Keywords

• nonn

Comments

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.

Examples

			The sequence of terms together with the corresponding compositions begins:
    0: ()               127: (1,1,1,1,1,1,1)
    1: (1)              149: (3,2,2,1)
    3: (1,1)            151: (3,2,1,1,1)
    5: (2,1)            171: (2,2,2,1,1)
    7: (1,1,1)          175: (2,2,1,1,1,1)
   11: (2,1,1)          191: (2,1,1,1,1,1,1)
   15: (1,1,1,1)        255: (1,1,1,1,1,1,1,1)
   21: (2,2,1)          293: (3,3,2,1)
   23: (2,1,1,1)        299: (3,2,2,1,1)
   31: (1,1,1,1,1)      303: (3,2,1,1,1,1)
   37: (3,2,1)          341: (2,2,2,2,1)
   43: (2,2,1,1)        343: (2,2,2,1,1,1)
   47: (2,1,1,1,1)      351: (2,2,1,1,1,1,1)
   63: (1,1,1,1,1,1)    383: (2,1,1,1,1,1,1,1)
   75: (3,2,1,1)        511: (1,1,1,1,1,1,1,1,1)
   85: (2,2,2,1)        549: (4,3,2,1)
   87: (2,2,1,1,1)      587: (3,3,2,1,1)
   95: (2,1,1,1,1,1)    597: (3,2,2,2,1)
		

Crossrefs

Sequences covering an initial interval are counted by A000670.

Compositions in standard order are A066099.

Weakly decreasing runs are counted by A124765.

Removing the covering condition gives A114994.

Removing the ordering condition gives A333217.

The strictly decreasing case is A246534.

The unequal version is A333218.

The weakly increasing version is A333379.

Cf. A000120, A000225, A029931, A048793, A070939, A164894, A225620, A228351, A233564, A272919, A333219.

Programs

Mathematica

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],normQ[stc[#]]&&GreaterEqual@@stc[#]&]

Formula

Intersection of A333217 and A114994.