A374355 -id:A374355 - cp's OEIS frontend

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

Rémy Sigrist, Jul 06 2024

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)

			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

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

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.

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 *)

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); }

a(n) = A374354(n, A277561(n)-1).
a(n) = n - A374355(n).
a(n) <= n with equality iff n is a fibbinary number.

A374354 Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers 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).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 4, 0, 1, 4, 5, 2, 4, 2, 5, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 1, 2, 9, 10, 4, 8, 4, 5, 8, 9, 4, 10, 5, 10, 0, 16, 0, 1, 16, 17, 0, 2, 16, 18, 1, 2, 17, 18, 0, 4, 16, 20, 0, 1, 4, 5, 16, 17, 20, 21, 2, 4, 18, 20, 2, 5, 18, 21, 8, 16, 8, 9, 16, 17

Offset: 0

Rémy Sigrist, Jul 06 2024

In other words, we partition n into pairs of fibbinary numbers whose binary expansions have no common 1's and list the corresponding fibbinary numbers to get the n-th row.

			Triangle T(n, k) begins:
  n   n-th row
  --  -----------
   0  0
   1  0, 1
   2  0, 2
   3  1, 2
   4  0, 4
   5  0, 1, 4, 5
   6  2, 4
   7  2, 5
   8  0, 8
   9  0, 1, 8, 9
  10  0, 2, 8, 10
  11  1, 2, 9, 10
  12  4, 8
  13  4, 5, 8, 9
  14  4, 10
  15  5, 10
  16  0, 16

Rémy Sigrist, Table of n, a(n) for n = 0..8118 (rows for n = 0..1023 flattened)
Index entries for sequences related to binary expansion of n

See A295989 and A374361 for similar sequences.

Cf. A003714, A037800, A277561, A374355, A374356.

row(n) = { my (r = [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], r = concat([v + y | v <- r], [v + x | v <- r]); break;););); return (r); }

T(n, 0) = 0 iff n is a fibbinary number.
T(n, k) + T(n, A277561(n)-1-k) = n.
T(n, 0) = A374355(n).
T(n, A277561(n)-1) = A374356(n).
Sum_{k = 0..A277561(n)-1} T(n, k) = n * 2^A037800(n).

A374395 a(n) is the first term in the n-th row of A374394.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 0, 1, 3, 3, 4, 0, 2, 2, 5, 6, 5, 7, 7, 0, 1, 3, 3, 4, 8, 10, 10, 8, 9, 11, 11, 12, 0, 2, 2, 5, 6, 5, 7, 7, 13, 14, 16, 16, 17, 13, 15, 15, 18, 19, 18, 20, 20, 0, 1, 3, 3, 4, 8, 10, 10, 8, 9, 11, 11, 12, 21, 23, 23, 26, 27, 26, 28, 28, 21

Offset: 0

Rémy Sigrist, Jul 10 2024

a(n) is the least number z >= 0 such that the Zeckendorf representations of z and n-z have no common Fibonacci numbers and when combined together correspond to the lazy Fibonacci representation of n.

Rémy Sigrist, Table of n, a(n) for n = 0..10944
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003754, A022290, A374355, A374394, A374396.

PARI
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a(n) = A022290(A374355(1+A003754(n)), k).

a(n) = n - A374396(n).

A377415 a(n) = n - A377414(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 2, 0, 1, 0, 1, 4, 5, 4, 5, 0, 0, 2, 2, 0, 0, 2, 2, 8, 8, 10, 10, 8, 8, 10, 10, 0, 1, 0, 1, 4, 5, 4, 5, 0, 1, 0, 1, 4, 5, 4, 5, 16, 17, 16, 17, 20, 21, 20, 21, 16, 17, 16, 17, 20, 21, 20, 21, 0, 0, 2, 2, 0, 0, 2, 2, 8, 8, 10, 10, 8, 8, 10

Offset: 0

Rémy Sigrist, Oct 27 2024

For any n > 0 with binary expansion (b_1 = 1, b_2, ..., b_k), the binary expansion of a(n) is (c_1, ..., c_k) where c_i = b_i when i is even, c_i = 0 when i is odd.

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

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

See A063694, A063695 and A374355 for similar sequences.

Cf. A126684, A371459, A377414.

a(n) = { my (v = 0, x = exponent(n), y); while (n, n -= 2^y = exponent(n); if (x%2 != y%2, v += 2^y;);); return (v); }

a(n) = 0 iff n belongs to A126684.
a(a(n)) = 0.
a(2*n) = 2*a(n).

cp's OEIS Frontend

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).

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A374354 Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers 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).

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A374395 a(n) is the first term in the n-th row of A374394.

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A377415 a(n) = n - A377414(n).

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