A390745 Numbers k such that there exists a unique integer composition whose first sums are the k-th composition in standard order.

2, 10, 18, 20, 34, 40, 42, 66, 74, 80, 84, 130, 138, 146, 148, 160, 164, 168, 170, 258, 266, 274, 276, 296, 298, 320, 328, 336, 340, 514, 522, 530, 532, 546, 552, 554, 580, 586, 592, 596, 640, 648, 656, 658, 672, 676, 680, 682, 1026, 1034, 1042, 1044, 1058

Offset

1,1

Comments

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..53.

Example

For n = 68, the composition (4,3) is the first sums of (2,2,1) and (3,1,2), so 68 is not in the sequence.
For n = 164, composition (2,3,3) is the first sums of (1,1,2,1) only, so 164 is in the sequence.
The terms together with standard compositions begin:
    2: (2)
   10: (2,2)
   18: (3,2)
   20: (2,3)
   34: (4,2)
   40: (2,4)
   42: (2,2,2)
   66: (5,2)
   74: (3,2,2)
   80: (2,5)
   84: (2,2,3)
  130: (6,2)
  138: (4,2,2)
  146: (3,3,2)
  148: (3,2,3)
  160: (2,6)
  164: (2,3,3)
  168: (2,2,4)
  170: (2,2,2,2)

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&]]==1&]

Crossrefs

These indicate terms appearing only once in A390568.

These are positions of 1 in the pre-bisected A390675.

The not necessarily unique version is A390676, complement A390677.

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 distinct first sums, counted by A390567.

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

Cf. A000079, A000120, A001511, A008965, A029837, A029931, A070211, A070939, A357135, A390448, A390678.

Sequence in context: A341032 A082939 A074069 * A069081 A126005 A017089

Adjacent sequences: A390742 A390743 A390744 * A390746 A390747 A390748

Keyword

nonn

Author

Gus Wiseman, Nov 22 2025

Status

approved