A384879 Numbers whose binary indices have all distinct lengths of maximal anti-runs (increasing by more than 1).

1, 2, 4, 5, 8, 9, 10, 11, 13, 16, 17, 18, 19, 20, 21, 22, 25, 26, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 49, 50, 52, 53, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 88, 97, 98, 100, 101, 104, 105, 106, 128, 129, 130

Offset

1,2

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.

Links

Table of n, a(n) for n=1..64.

Example

The binary indices of 813 are {1,3,4,6,9,10}, with maximal anti-runs ((1,3),(4,6,9),(10)), with lengths (2,3,1), so 813 is in the sequence.
The terms together with their binary expansions and binary indices begin:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    5:     101 ~ {1,3}
    8:    1000 ~ {4}
    9:    1001 ~ {1,4}
   10:    1010 ~ {2,4}
   11:    1011 ~ {1,2,4}
   13:    1101 ~ {1,3,4}
   16:   10000 ~ {5}
   17:   10001 ~ {1,5}
   18:   10010 ~ {2,5}
   19:   10011 ~ {1,2,5}
   20:   10100 ~ {3,5}
   21:   10101 ~ {1,3,5}
   22:   10110 ~ {2,3,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[100], UnsameQ@@Length/@Split[bpe[#], #2!=#1+1&]&]

Crossrefs

Subsets of this type are counted by A384177, for runs A384175 (complement A384176).

These are the indices of strict rows in A384877, see A384878, A245563, A245562, A246029.

A000120 counts binary indices.

A098859 counts Wilf partitions (distinct multiplicities), complement A336866.

A356606 counts strict partitions without a neighborless part, complement A356607.

A384890 counts maximal anti-runs in binary indices, runs A069010.

Cf. A023758, A044813, A048793, A164707, A242882, A243815, A325325, A328592, A384879, A384884, A384886, A384893.

Sequence in context: A189633 A032891 A377169 * A035262 A010440 A072008

Adjacent sequences: A384876 A384877 A384878 * A384880 A384881 A384882

Keyword

nonn

Author

Gus Wiseman, Jun 17 2025

Status

approved