A374356 - OEIS

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

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OFFSET

0,3

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)

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Index entries for sequences related to binary expansion of n

FORMULA

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

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

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

EXAMPLE

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

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 *)

PROG

(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); }

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.

Sequence in context: A292271 A292593 A332995 * A214793 A199088 A293974

Adjacent sequences: A374353 A374354 A374355 * A374357 A374358 A374359

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Jul 06 2024

STATUS

approved