A381852 In the binary expansion of n (without leading zeros): complement the bits strictly to the right of the leftmost zero digit, if any.

0, 1, 2, 3, 5, 4, 6, 7, 11, 10, 9, 8, 13, 12, 14, 15, 23, 22, 21, 20, 19, 18, 17, 16, 27, 26, 25, 24, 29, 28, 30, 31, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 55, 54, 53, 52, 51, 50, 49, 48, 59, 58, 57, 56, 61, 60, 62, 63, 95, 94, 93, 92

Offset

0,3

Comments

This sequence is a self-inverse permutation of the nonnegative integers.

This sequence has similarities with A054429 (where we complement the bits to the right of the leftmost one digit).

Links

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

Rémy Sigrist, Colored scatterplot of the sequence for n = 0..2^13-1 (where the color denotes the position of the leftmost zero digit in the binary expansion of n)

Index entries for sequences related to binary expansion of n

Index entries for sequences that are permutations of the natural numbers

Formula

a(n) = n iff n = 0 or n belongs to A075427.

a(n) = XOR(n,2^(A063250(n)-1)-1) if n>0 and A063250(n)>0. Otherwise a(n) = n. - Chai Wah Wu, Mar 09 2025

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     3      11         11
   4     5     100        101
   5     4     101        100
   6     6     110        110
   7     7     111        111
   8    11    1000       1011
   9    10    1001       1010
  10     9    1010       1001
  11     8    1011       1000
  12    13    1100       1101
  13    12    1101       1100
  14    14    1110       1110
  15    15    1111       1111
  16    23   10000      10111

Prog

(PARI) a(n) = { my (b = binary(n)); for (i = 1, #b, if (b[i]==0, for (j = i+1, #b, b[j] = 1-b[j]; ); return (fromdigits(b, 2)); ); ); return (n); }
(Python)
def a(n):
    b = bin(n)[2:]
    zi = b.find('0')
    return n if zi == -1 else int(b[:zi+1]+"".join('0' if bi == '1' else '1' for bi in b[zi+1:]), 2)
print([a(n) for n in range(70)]) # Michael S. Branicky, Mar 09 2025
(Python)
def A381852(n): return n^((1<<n.bit_length()-t-1)-1) if n and (t:=bin(n)[2:].find('0'))!=-1 else n # Chai Wah Wu, Mar 09 2025

Crossrefs

Cf. A054429, A063250, A075427, A381839.

Sequence in context: A069766 A081241 A180201 * A233277 A279339 A279349

Adjacent sequences: A381849 A381850 A381851 * A381853 A381854 A381855

Keyword

nonn,base,easy

Author

Rémy Sigrist, Mar 08 2025

Status

approved