A390677 Numbers k such that there is no integer composition whose first sums are the k-th composition in standard order.

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86

Offset

1,2

Comments

First differs from A356844 in having 82, corresponding to the composition (2,3,2).

The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

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.

Links

Table of n, a(n) for n=1..67.

Example

For 72 we have the standard composition (3,4), which is the first sums of two compositions: (1,2,2) and (2,1,3), so 72 is not in the sequence.
For 82 we have the standard composition (2,3,2), which is not the first sums of any composition, so 82 is in the sequence.
For 658 we have the standard composition (2,3,3,2), which is the first sums of (1,1,2,1,1), so 658 is not in the sequence.
The terms together with standard compositions begin:
   1: (1)
   3: (1,1)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   9: (3,1)
  11: (2,1,1)
  12: (1,3)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  17: (4,1)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
uncha[tar_, ini_]:=(cur={ini}; Do[AppendTo[cur, tar[[k]]-If[k==1, ini, cur[[k]]]], {k, Length[tar]}]; cur);
Select[Range[100], Length[Select[Table[uncha[stc[#], i], {i, Max[stc[#]]}], Min@@#>=1&]]==0&]

Crossrefs

These indicate zeros in the pre-bisected A390675.

Complement of A390676 = union of A390568.

The unique version is A390745.

A011782 counts compositions.

A066099 lists all compositions in standard order.

A357213 counts compositions by sum of first sums.

A390432 lists first sums of standard compositions.

A390449 ranks first sums of prime indices, listed by A390307 or A390362.

A390673 ranks compositions with all distinct first sums, counted by A390567.

A390674 ranks compositions with all equal first sums, counted by A342527.

Cf. A000079, A000120, A001511, A008965, A029837, A029931, A070211, A070939, A389731, A390448, A390678.

Sequence in context: A333227 A285510 A174415 * A356844 A103826 A361386

Adjacent sequences: A390674 A390675 A390676 * A390678 A390679 A390680

Keyword

nonn

Author

Gus Wiseman, Nov 23 2025

Status

approved