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A035327 Write n in binary, interchange 0's and 1's, convert back to decimal. 74
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
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
For n >= 2, a(n) is the least nonnegative k that must be added to n+1 to make a power of 2. Hence in a single-elimination tennis tournament with n entrants, a(n-1) is the number of players given a bye in round one, so that the number of players remaining at the start of round two is a power of 2. For example, if 39 players register, a(38)=25 players receive a round-one bye leaving 14 to play, so that round two will have 25+(14/2)=32 players. - Mathew Englander, Jan 20 2024
LINKS
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
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
a(n) = abs(2*A053644(n) - n - 1). - Mathew Englander, Jan 22 2024
EXAMPLE
8 = 1000 -> 0111 = 111 = 7.
MAPLE
seq(2^(1 + ilog2(max(n, 1))) - 1 - n, n = 0..81); # Emeric Deutsch, Oct 19 2008
A035327 := n -> `if`(n=0, 1, Bits:-Nand(n, n)):
seq(A035327(n), n=0..81); # Peter Luschny, Sep 23 2019
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 *)
Join[{1}, Table[2^BitLength[n]-n-1, {n, 100}]] (* Paolo Xausa, Oct 13 2023 *)
PROG
(PARI) a(n)=sum(k=1, n, if(bitxor(n, k)>n, 1, 0)) \\ Paul D. Hanna, Jan 21 2006
(PARI) a(n) = bitxor(n, 2^(1+logint(max(n, 1), 2))-1) \\ Rémy Sigrist, Jan 04 2019
(PARI) a(n)=if(n, bitneg(n, exponent(n)+1), 1) \\ Charles R Greathouse IV, Apr 13 2020
(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
(Python)
def a(n): return 1 if n == 0 else n^((1 << n.bit_length()) - 1)
print([a(n) for n in range(100)]) # Michael S. Branicky, Sep 28 2021
(Python)
def A035327(n): return (~n)^(-1<<n.bit_length()) if n else 1 # Chai Wah Wu, Dec 20 2022
(SageMath)
def a(n):
if n == 0:
return 1
return sum([(1 - b) << s for (s, b) in enumerate(n.bits())])
[a(n) for n in srange(82)] # Peter Luschny, Aug 31 2019
(Julia)
using IntegerSequences
A035327List(len) = [Bits("NAND", n, n) for n in 0:len]
println(A035327List(100)) # Peter Luschny, Sep 25 2021
CROSSREFS
a(n) = A003817(n) - n, for n>0.
Cf. A240857.
Sequence in context: A323908 A098825 A111460 * A004444 A204533 A357079
KEYWORD
nonn,easy,base,look
AUTHOR
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 June 13 09:03 EDT 2024. Contains 373383 sequences. (Running on oeis4.)