A357708 Numbers k such that the k-th composition in standard order has sum equal to twice its maximum part.

3, 10, 11, 13, 14, 36, 37, 38, 39, 41, 44, 50, 51, 52, 57, 60, 136, 137, 138, 139, 140, 141, 142, 143, 145, 152, 162, 163, 168, 177, 184, 196, 197, 198, 199, 200, 209, 216, 226, 227, 232, 241, 248, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539

Offset: 1

Author

Gus Wiseman, Oct 14 2022

Keywords

• nonn

Comments

The k-th composition in standard order (graded reverse-lexicographic, 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. This gives a bijective correspondence between nonnegative integers and integer compositions.

Examples

			The terms and corresponding compositions begin:
    3: (1,1)
   10: (2,2)
   11: (2,1,1)
   13: (1,2,1)
   14: (1,1,2)
   36: (3,3)
   37: (3,2,1)
   38: (3,1,2)
   39: (3,1,1,1)
   41: (2,3,1)
   44: (2,1,3)
   50: (1,3,2)
   51: (1,3,1,1)
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

See link for sequences related to standard compositions.

A066311 lists gapless numbers.

A124767 counts runs in standard compositions.

A333766 gives maximal part of standard compositions, minimal A333768.

A356844 ranks compositions with at least one 1.

Cf. A000120, A001511, A003754, A029931, A329395.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Max@@stc[#]==Total[stc[#]]/2&]