A374356 a(n) is the greatest fibbinary number f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

0, 1, 2, 2, 4, 5, 4, 5, 8, 9, 10, 10, 8, 9, 10, 10, 16, 17, 18, 18, 20, 21, 20, 21, 16, 17, 18, 18, 20, 21, 20, 21, 32, 33, 34, 34, 36, 37, 36, 37, 40, 41, 42, 42, 40, 41, 42, 42, 32, 33, 34, 34, 36, 37, 36, 37, 40, 41, 42, 42, 40, 41, 42, 42, 64, 65, 66, 66

Offset: 0

Author

Rémy Sigrist, Jul 06 2024

Keywords

• nonn
• base

Comments

To compute a(n): replace every other bit with zero (starting with the second bit) in each run of consecutive 1's in the binary expansion of n.

From Gus Wiseman, Jul 11 2025: (Start)

This is the greatest binary rank of a sparse subset of the binary indices of n, where:

1. The binary indices of a nonnegative integer are the positions of 1 in its reversed binary expansion.

2. A set is sparse iff 1 is not a first difference.

3. The binary rank of a set {S_1,S_2,...} is Sum_i 2^(S_i-1).

(End)

Examples

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2      10         10
   3     2      11         10
   4     4     100        100
   5     5     101        101
   6     4     110        100
   7     5     111        101
   8     8    1000       1000
   9     9    1001       1001
  10    10    1010       1010
  11    10    1011       1010
  12     8    1100       1000
  13     9    1101       1001
  14    10    1110       1010
  15    10    1111       1010
  16    16   10000      10000
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8191
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A277561, A374354, A374355.

The union is A003714 (Fibbinary numbers).

For prime instead of binary indices we have A385216.

A034839 counts subsets by number of maximal runs, for strict partitions A116674.

A166469 counts sparse submultisets of prime indices, maximal A385215.

A245564 counts sparse subsets of binary indices, maximal case A384883.

A319630 ranks sparse submultisets of prime indices, complement A104210.

Cf. A000045, A000071, A001629, A006519, A010049, A044813, A119900, A202023, A202064, A268193, A384890.

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fbi[q_]:=If[q=={},0,Total[2^q]/2];
Table[Max@@fbi/@Select[Subsets[bpe[n]],FreeQ[Differences[#],1]&],{n,0,100}] (* Gus Wiseman, Jul 11 2025 *)

PARI

a(n) = { my (v = 0, e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], v += x; break;););); return (v); }

Formula

a(n) = A374354(n, A277561(n)-1).

a(n) = n - A374355(n).

a(n) <= n with equality iff n is a fibbinary number.