Search: id:A035327
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%I A035327
%S A035327 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,
%T A035327 31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,
%U A035327 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
%N A035327 Write n in binary, interchange 0's and 1's.
%C A035327 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 Delarte (alonso.delarte(AT)gmail.com), Jan 14 2006
%C A035327 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. [From Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 14 2009]
%D A035327 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. 
               Computer Sci., 307 (2003), 3-29.
%H A035327 R. Zumkeller, <a href="b035327.txt">Table of n, a(n) for n = 0..10000</
               a> [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 
               14 2009]
%H A035327 J.-P. Allouche and J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">
               The Ring of k-regular Sequences, II</a>
%H A035327 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer 
               generating functions. I. Elementary sequences</a>
%H A035327 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A035327 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%F A035327 a(n) = 2^k - n - 1, where 2^(k-1) < n < 2^k.
%F A035327 a(n+1) = (a(n)+n) mod (n+1); a(0) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jul 22 2002
%F A035327 G.f.: 1 + 1/(1-x)*sum(k>=0, 2^k*x^2^(k+1)/(1+x^2^k)). - Ralf Stephan, 
               May 06 2003
%F A035327 a(0) = 0, a(2n+1) = 2*a(n), a(2n) = 2*a(n) + 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Feb 29 2004
%F A035327 a(n) = number of positive integers k < n such that n XOR k > n. a(n) 
               = n - A006257(n). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 21 
               2006
%F A035327 a(n)=2^{1+floor(log[2](n))}-n-1 for n>=1; a(0)=1. [From Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Oct 19 2008]
%e A035327 8 = 1000 -> 0111 = 111 = 7
%p A035327 1,seq(2^(1+floor(log[2](n)))-n-1,n=1..81); [From Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Oct 19 2008]
%t A035327 Table[BaseForm[FromDigits[(IntegerDigits[i, 2]/.{0->1, 1->0}), 2], 10], 
               {i, 0, 90}]
%t A035327 Table[BitXor[n, 2^IntegerPart[Log[2, n] + 1] - 1], {n, 100}] (Delarte)
%o A035327 (PARI) a(n)=sum(k=1,n,if(bitxor(n,k)>n,1,0)) (Hanna)
%Y A035327 a(n) = A003817(n) - n, for n>0. Cf. A087734.
%Y A035327 Cf. A000225, A006257 (Josephus problem).
%Y A035327 Sequence in context: A051427 A098825 A111460 this_sequence A004444 A085771 
               A111106
%Y A035327 Cf. A167831, A167877. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 14 2009]
%Y A035327 Adjacent sequences: A035324 A035325 A035326 this_sequence A035328 A035329 
               A035330
%K A035327 nonn,easy,base
%O A035327 0,5
%A A035327 N. J. A. Sloane (njas(AT)research.att.com).
%E A035327 More terms from Vit Planocka (planocka(AT)mistral.cz), Feb 01 2003
%E A035327 a(0) corrected by Paolo P. Lava (ppl(AT)spl.at), Oct 22 2007

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