A241816 a(n) is the largest number smaller than n that can be obtained by swapping two adjacent bits in n, or n if no such number exists.

0, 1, 1, 3, 2, 3, 5, 7, 4, 5, 9, 7, 10, 11, 13, 15, 8, 9, 17, 11, 18, 19, 21, 15, 20, 21, 25, 23, 26, 27, 29, 31, 16, 17, 33, 19, 34, 35, 37, 23, 36, 37, 41, 39, 42, 43, 45, 31, 40, 41, 49, 43, 50, 51, 53, 47, 52, 53, 57, 55, 58, 59, 61, 63, 32, 33, 65, 35, 66, 67, 69

Offset

0,4

Comments

Equivalently, a(n) is obtained by swapping the first pair of adjacent bits equal to "10", if such a pair exists. [Here the first pair means the first pair from the right, the least significant end of binary expansion. Comment clarified by Antti Karttunen, Feb 20 2015.]

The fixed point of a(n) is equal to 2^k - 1, where k = A000120(n). In other words, applying a(n) repeatedly packs all the bits to the right.

a(n) is related to the "bubble sort" algorithm. If an array of elements from two classes is encoded in a binary number, a(n) is the first intermediate result that will be obtained when starting a bubble sort from n.

Links

Philippe Beaudoin, Table of n, a(n) for n = 0..9999

Formula

a(0) = 0; a(2n+1) = 1+2*a(n), a(4n) = 2*a(2n), a(4n+2) = 4n+1. - Antti Karttunen, Feb 21 2015

Example

If n = 5 = 101_2 then a(n) = 011_2 = 3.
If n = 8 = 1000_2 then a(8) = 0100_2 = 4.

Prog

(Python)
def bitswap(n):
..# Find first bit = 0.
..m = n
..i = 0
..while (m > 0):
....if m % 2 == 0:
......break
....m = m >> 1
....i = i + 1
..if m == 0:
.....return n
..# Find first bit = 1 following that 0.
..while (m > 0):
....if m % 2 == 1:
......break
....m = m >> 1
....i = i + 1
..# Swap
..return n & ~(1 << i) | (1 << (i-1))
(Haskell)
a241816 n = f (a030308_row n) [] where
   f [] _ = n
   f (0 : 1 : us) vs = foldr (\b y -> 2 * y + b) 0 $
                             reverse vs ++ 1 : 0 : us
   f (u : us) vs     = f us (u : vs)
-- Reinhard Zumkeller, Sep 03 2014
(Python)
def A241816(n):
....s = bin(n)[2:]
....for i in range(len(s)-2, -1, -1):
........if s[i:i+2] == '10':
............return int(s[:i]+'01'+s[i+2:], 2)
....else:
........return n
# Chai Wah Wu, Sep 05 2014
(Scheme, with memoization-macro definec)
(definec (A241816 n) (cond ((zero? n) n) ((odd? n) (+ 1 (* 2 (A241816 (/ (- n 1) 2))))) ((zero? (modulo n 4)) (* 2 (A241816 (/ n 2)))) (else (- n 1))))
;; Antti Karttunen, Feb 21 2015

Crossrefs

Cf. A055941, A030101, A030308, A007088, A246591, A246592, A246593, A246594.

Complement is A246590.

Sequence in context: A247829 A351705 A246592 * A243109 A206012 A211940

Adjacent sequences: A241813 A241814 A241815 * A241817 A241818 A241819

Keyword

nonn,base

Author

Philippe Beaudoin, Aug 19 2014

Extensions

Definition clarified by Chai Wah Wu, Sep 05 2014

Status

approved