A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.

1, 2, 4, 8, 11, 13, 16, 19, 22, 25, 26, 32, 35, 38, 44, 49, 50, 52, 64, 67, 70, 76, 87, 88, 91, 93, 97, 98, 100, 104, 107, 109, 117, 128, 131, 134, 140, 151, 152, 155, 157, 167, 174, 176, 179, 182, 185, 186, 193, 194, 196, 200, 203, 205, 208, 211, 214, 217

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..58.

Example

The terms together with their binary expansions and binary indices begin:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    8:    1000 ~ {4}
   11:    1011 ~ {1,2,4}
   13:    1101 ~ {1,3,4}
   16:   10000 ~ {5}
   19:   10011 ~ {1,2,5}
   22:   10110 ~ {2,3,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}
   32:  100000 ~ {6}
   35:  100011 ~ {1,2,6}
   38:  100110 ~ {2,3,6}
   44:  101100 ~ {3,4,6}
   49:  110001 ~ {1,5,6}
   50:  110010 ~ {2,5,6}
   52:  110100 ~ {3,5,6}
   64: 1000000 ~ {7}
   67: 1000011 ~ {1,2,7}
   70: 1000110 ~ {2,3,7}
   76: 1001100 ~ {3,4,7}
   87: 1010111 ~ {1,2,3,5,7}
   88: 1011000 ~ {4,5,7}
   91: 1011011 ~ {1,2,4,5,7}
   93: 1011101 ~ {1,3,4,5,7}
   97: 1100001 ~ {1,6,7}
   98: 1100010 ~ {2,6,7}
  100: 1100100 ~ {3,6,7}

Mathematica

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

Prog

(PARI) is_ok(n)=hammingweight(n)==2*hammingweight(bitand(n, n>>1))+1 \\ Christian Sievers, Jul 18 2025

Crossrefs

The LHS rank statistic is A069010, counted by A034839 (for partitions A384881, A116674).

The RHS rank statistic is A384890, counted by A384893 (for partitions A268193, A384905).

Subsets of this type are counted by A385572, with n A217615.

A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.

A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.

Cf. A000079, A010027, A053538, A210034, A384177, A384879, A384889.

Sequence in context: A078649 A161607 A318206 * A022442 A308163 A056689

Adjacent sequences: A385572 A385573 A385574 * A385576 A385577 A385578

Keyword

nonn

Author

Gus Wiseman, Jul 04 2025

Status

approved