A390091 Numbers whose binary indices have distinct first differences.

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 22, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 40, 41, 44, 48, 49, 50, 52, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 80, 81, 82, 83, 88, 89, 96, 97, 98, 100, 101, 104, 105, 128, 129, 130, 131, 132

Offset

1,3

Comments

First differs from A337582 in having 75.

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. This sequence also works for positions of 1's in binary expansion (not reversed).

Links

Table of n, a(n) for n=1..66.

Example

The binary indices of 75 are (1,2,4,7), with differences (1,2,3), so 75 is in the sequence.
The terms together with their binary expansions and binary indices begin:
    0:       0 ~ {}
    1:       1 ~ {1}
    2:      10 ~ {2}
    3:      11 ~ {1,2}
    4:     100 ~ {3}
    5:     101 ~ {1,3}
    6:     110 ~ {2,3}
    8:    1000 ~ {4}
    9:    1001 ~ {1,4}
   10:    1010 ~ {2,4}
   11:    1011 ~ {1,2,4}
   12:    1100 ~ {3,4}
   13:    1101 ~ {1,3,4}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 100], UnsameQ@@Differences[bpe[#]]&]

Crossrefs

For sums of subsets we have A059519, counted by A275972 and A325864.

The prepended version is A233564.

For equal instead of distinct differences we have A295235, counted by A175342.

Partitions of this type are counted by A325325, ranks A325368.

Compositions of this type are counted by A325545, ranks A389597.

For all increasing pairs (not just adjacent) we have A337582.

Subsets of this type are counted by A364465.

Indicates strict rows of A390090.

For strictly increasing differences we have A390305, decreasing A390306.

Cf. A000120, A000740, A001511, A029837, A030101, A070939, A325546, A325551, A325554, A329739, A389601, A389734.

Sequence in context: A247063 A003726 A343110 * A337582 A004828 A032895

Adjacent sequences: A390088 A390089 A390090 * A390092 A390093 A390094

Keyword

nonn

Author

Gus Wiseman, Nov 19 2025

Status

approved