A390674 - OEIS

A390674

Numbers k > 0 such that the k-th composition in standard order has all equal first sums.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 31, 32, 33, 34, 36, 40, 42, 45, 48, 49, 54, 63, 64, 65, 66, 68, 72, 76, 80, 82, 96, 97, 109, 127, 128, 129, 130, 132, 136, 144, 148, 153, 160, 162, 170, 182, 192, 193, 204, 255, 256, 257

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OFFSET

1,2

COMMENTS

First differs from A353744 in lacking 37, corresponding to the composition (3,2,1).

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.

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

LINKS

Table of n, a(n) for n=1..62.

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

EXAMPLE

The terms together with standard compositions begin:

1: (1)

2: (2)

3: (1,1)

4: (3)

5: (2,1)

6: (1,2)

7: (1,1,1)

8: (4)

9: (3,1)

10: (2,2)

12: (1,3)

13: (1,2,1)

15: (1,1,1,1)

MATHEMATICA

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

firsums[c_]:=Table[c[[i]]+c[[i+1]], {i, Length[c]-1}];

Select[Range[100], SameQ@@firsums[stc[#]]&]

CROSSREFS

These compositions are counted by A342527.

The version for Heinz numbers of partitions is A354144, counted by A361003.

For first differences instead of sums we have A389731, counted by A175342.

These are the positions of constant rows in A390432.

For distinct instead of equal first sums we have A390673, counted by A390567.

A011782 counts compositions.

A066099 lists all compositions in standard order.

A351014 counts distinct runs in standard compositions.

A390307 lists first sums of prime indices, ranks A390449.

Cf. A000120, A001511, A029837, A029931, A070939, A325545, A357135, A358133, A358134, A358135, A390568.

Sequence in context: A354581 A193096 A353744 * A309129 A307345 A033110

Adjacent sequences: A390671 A390672 A390673 * A390675 A390676 A390677

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 17 2025

STATUS

approved