A385889 The number k such that the k-th composition in standard order is the sequence of lengths of maximal runs of binary indices of n.

0, 1, 1, 2, 1, 3, 2, 4, 1, 3, 3, 5, 2, 6, 4, 8, 1, 3, 3, 5, 3, 7, 5, 9, 2, 6, 6, 10, 4, 12, 8, 16, 1, 3, 3, 5, 3, 7, 5, 9, 3, 7, 7, 11, 5, 13, 9, 17, 2, 6, 6, 10, 6, 14, 10, 18, 4, 12, 12, 20, 8, 24, 16, 32, 1, 3, 3, 5, 3, 7, 5, 9, 3, 7, 7, 11, 5, 13, 9, 17, 3

Offset

0,4

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

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

John Tyler Rascoe, Table of n, a(n) for n = 0..8192

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

John Tyler Rascoe, Python code.

Example

The binary indices of 27 are {1,2,4,5}, with maximal runs ((1,2),(4,5)), with lengths (2,2), which is the 10th composition in standard order, so a(27) = 10.
The binary indices of 100 are {3,6,7}, with maximal runs ((3),(6,7)), with lengths (1,2), which is the 6th composition in standard order, so a(100) = 6.

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Length/@Split[bpe[n], #2==#1+1&]], {n, 0, 100}]

Prog

(Python) # see links

Crossrefs

Sorted positions of firsts appearances appear to be A247648+1.

After removing duplicates we get A385818.

The reverse version is A385887.

A245563 lists run lengths of binary indices (ranks A246029), reverse A245562.

A384877 lists anti-run lengths of binary indices (ranks A385816), reverse A209859.

Cf. A000120, A023758, A029931, A044813, A048793, A069010, A348366, A384175, A384878, A385817.

Sequence in context: A349191 A336394 A336472 * A385887 A058933 A087470

Adjacent sequences: A385886 A385887 A385888 * A385890 A385891 A385892

Keyword

nonn,easy

Author

Gus Wiseman, Jul 16 2025

Status

approved