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A035327 Write n in binary, interchange 0's and 1's, convert back to decimal. 30
1, 0, 1, 0, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Also bitwise XOR of n with the nearest Mersenne number (A000225) larger than or equal to n, for n > 0. (For n = 0, a(0) = -1 as opposed to 1). The advantage of using BitXor instead of BaseForm in the Mathematica program is that the result has a Head of Integer, not BaseForm. - Alonso del Arte, Jan 14 2006

For n>0: largest m<=n such that no carry occurs when adding m to n in binary arithmetic: A003817(n+1) = a(n) + n = a(n) XOR n. - Reinhard Zumkeller, Nov 14 2009

a(0) could be considered to be 0 (it was set so from 2004 to 2008) if the binary representation of zero was chosen to be the empty string. - Jason Kimberley, Sep 19 2011

For n > 0: A240857(n,a(n)) = 0. - Reinhard Zumkeller, Apr 14 2014

This is a base-2 analog of A048379. Another variant, without converting back to decimal, is given in A256078. - M. F. Hasler, Mar 22 2015

LINKS

R. Zumkeller, Table of n, a(n) for n = 0..10000

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

Index entries for sequences related to binary expansion of n

FORMULA

a(n) = 2^k - n - 1, where 2^(k-1) < n < 2^k.

a(n+1) = (a(n)+n) mod (n+1); a(0) = 1. - Reinhard Zumkeller, Jul 22 2002

G.f.: 1 + 1/(1-x)*sum(k>=0, 2^k*x^2^(k+1)/(1+x^2^k)). - Ralf Stephan, May 06 2003

a(0) = 0, a(2n+1) = 2*a(n), a(2n) = 2*a(n) + 1. - Philippe Deléham, Feb 29 2004

a(n) = number of positive integers k < n such that n XOR k > n. a(n) = n - A006257(n). - Paul D. Hanna, Jan 21 2006

a(n) = 2^{1+floor(log[2](n))}-n-1 for n>=1; a(0)=1. - Emeric Deutsch, Oct 19 2008

a(n) = if n<2 then 1 - n else 2*a(floor(n/2)) + 1 - n mod 2. - Reinhard Zumkeller, Jan 20 2010

EXAMPLE

8 = 1000 -> 0111 = 111 = 7.

MAPLE

1, seq(2^(1+floor(log[2](n)))-n-1, n=1..81); # Emeric Deutsch, Oct 19 2008

MATHEMATICA

Table[BaseForm[FromDigits[(IntegerDigits[i, 2]/.{0->1, 1->0}), 2], 10], {i, 0, 90}]

Table[BitXor[n, 2^IntegerPart[Log[2, n] + 1] - 1], {n, 100}] (* Alonso del Arte, Jan 14 2006 *)

PROG

(PARI) a(n)=sum(k=1, n, if(bitxor(n, k)>n, 1, 0)) \\ Paul D. Hanna, Jan 21 2006

(MAGMA) A035327:=func<n|n eq 0 select 1 else SequenceToInteger(([1-b:b in IntegerToSequence(n, 2)]), 2)>; // Jason Kimberley, Sep 19 2011

(Haskell)

a035327 n = if n <= 1 then 1 - n else 2 * a035327 n' + 1 - b

            where (n', b) = divMod n 2

-- Reinhard Zumkeller, Feb 21 2014

(Python)

def a(n): return int(''.join('1' if i == '0' else '0' for i in bin(n)[2:]), 2) # Indranil Ghosh, Apr 29 2017

CROSSREFS

a(n) = A003817(n) - n, for n>0. Cf. A087734.

Cf. A000225, A006257 (Josephus problem).

Cf. A167831, A167877, A007088, A061601, A171960, A010078.

Sequence in context: A194753 A098825 A111460 * A004444 A204533 A259790

Adjacent sequences:  A035324 A035325 A035326 * A035328 A035329 A035330

KEYWORD

nonn,easy,base,look

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vit Planocka (planocka(AT)mistral.cz), Feb 01 2003

a(0) corrected by Paolo P. Lava, Oct 22 2007

Definition completed by M. F. Hasler, Mar 22 2015

STATUS

approved

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Last modified February 23 09:46 EST 2018. Contains 299509 sequences. (Running on oeis4.)