A390673 - OEIS

A390673

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

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 24, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 48, 50, 52, 56, 58, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 78, 80, 81, 83, 84, 88, 92, 96, 98, 100, 101, 104, 105, 112, 114, 116

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OFFSET

1,3

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.

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

LINKS

John Tyler Rascoe, Table of n, a(n) for n = 1..8192

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

EXAMPLE

For 300 we have the standard composition (3,2,1,3), with first sums (5,3,4), which are distinct, so 300 is in the sequence.

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[0, 100], UnsameQ@@firsums[stc[#]]&]

CROSSREFS

For prime indices we have A004709, counted by A000726.