%I A007814
%S A007814 0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,
               2,0,
%T A007814 1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,
               0,1,
%U A007814 0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,
               1,0
%N A007814 Exponent of highest power of 2 dividing n (the binary carry sequence).
%C A007814 This sequence is an exception to my usual rule that when every other 
               term of a sequence is 0 then those 0's should be omitted. In this 
               case we would get A001511. - N. J. A. Sloane (njas(AT)research.att.com).
%C A007814 To construct the sequence: start with 0,1, concatenate to get 0,1,0,1. 
               Add + 1 to last term gives 0,1,0,2. Concatenate those 4 terms to 
               get 0,1,0,2,0,1,0,2. Add + 1 to last term etc. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Mar 06 2003
%C A007814 a(n) = A091090(n-1) + A036987(n-1) - 1.
%C A007814 Fixed point of the morphism 0->01, 1->02, 2->03, 3->04, ..., n->0(n+1), 
               ..., starting from a(1) = 0. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 15 2004
%C A007814 a(n) is also the number of times to repeat a step on an even number in 
               the hailstone sequence referenced in the Collatz conjecture. - Alex 
               T. Flood (whiteangelsgrace(AT)gmail.com), Sep 22 2006
%C A007814 Let F(n) be the n-th Fermat number (A000215). Then F(a(r-1)) divides 
               F(n)+2^k for r=mod(k,2^n) and r != 1. - T. D. Noe (noe(AT)sspectra.com), 
               Jul 12 2007
%C A007814 a(n) = A001511(n) - 1.
%C A007814 a(n) is the number of 0s at the end of n when n is written in base 2.
%C A007814 a(n) = A063787(n) - A000120(n) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jun 04 2009]
%D A007814 K. Atanassov, On the 37-th and the 38-th Smarandache Problems, Notes 
               on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 
               5 (1999), No. 2, 83-85.
%D A007814 K. Atanassov, On Some of Smarandache's Problems, American Research Press, 
               1999, 16-21.
%D A007814 Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps 
               Over the Natural Numbers (2009) http://arxiv.org/abs/0901.1397. Discrete 
               Math., 309 (2009), 6245-6254. [From njas, Nov 27 2009]
%D A007814 F. Smarandache, Only Problems, not Solutions!, Xiquan Publ., Phoenix-Chicago, 
               1993.
%D A007814 P. M. B. Vitanyi, An optimal simulation of counter machines, SIAM J. 
               Comput, 14:1(1985), 1-33.
%H A007814 T. D. Noe, <a href="b007814.txt">Table of n, a(n) for n=1..10000</a>
%H A007814 K. Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">
               On Some of Smarandache's Problems</a>
%H A007814 M. Hassani, <a href="http://jipam.vu.edu.au/article.php?sid=498">Equations 
               and inequalities involving v_p(n!)</a>, J. Inequ. Pure Appl. Math. 
               6 (2005) vol. 2, #29
%H A007814 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/
               ">Smarandache Notions Journal</a>
%H A007814 V. Shevelev, <a href="http://arXiv.org/abs/0904.2101">Several results 
               on sequences which are similar to the positive integers</a> [From 
               Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 15 2009]
%H A007814 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">
               Only Problems, Not Solutions!</a>.
%H A007814 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A007814 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%H A007814 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BinaryCarrySequence.html">Link to a section of The World of Mathematics.</
               a>
%H A007814 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Double-FreeSet.html">Link to a section of The World of Mathematics.</
               a>
%H A007814 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Binary.html">Binary</a>
%F A007814 a(n) = if n is odd then 0 else 1 + a(n/2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 11 2001
%F A007814 Sum(k=1, n, a(k))=n-A000120(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Oct 19 2002
%F A007814 G.f.: A(x) = Sum(k=1, infinity, x^(2^k)/(1-x^(2^k))). - Ralf Stephan 
               (ralf(AT)ark.in-berlin.de), Apr 10 2002
%F A007814 The sequence is invariant under the following two transformations: increment 
               every element by one (1, 2, 1, 3, 1, 2, 1, 4, ..), put a zero in 
               front and between adjacent elements (0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 
               0, 2, 0, 1, 0, 4, ..). The intermediate result is A001511. - Ralf 
               Hinze (ralf(AT)informatik.uni-bonn.de), Aug 26 2003
%F A007814 G.f. A(x) satisfies A(x) = A(x^2) + x^2/(1-x^2). A(x) = B(x^2) = B(x) 
               - x/(1-x), where B(x) is the g.f. for A001151. - Frank Adams-Watters 
               (FrankTAW(AT)Netscape.net), Feb 09 2006
%F A007814 Totally additive with a(p) = 1 if p = 2, 0 otherwise.
%F A007814 Dirichlet g.f.: zeta(s)/(2^s-1). - Ralf Stephan, Jun 17 2007
%F A007814 Define 0 <= k <= 2^n - 1; binary: k = b(0) + 2.b(1) + 4.b(2) + ... + 
               2^(n-1).b(n-1); where b(x) are 0 or 1 for 0 <= x <= n - 1; Define 
               c(x) = 1 - b(x) for 0 <= x <= n - 1; Then: a007814(k) = c(0) + c(0).c(1) 
               + c(0).c(1).c(2) + ... + c(0).c(1)...c(n-1); a007814(k+1) = b(0) 
               + b(0).b(1) + b(0).b(1).b(2) + ... + b(0).b(1)...b(n-1) - Arie Werksma 
               (werksma(AT)tiscali.nl), May 10 2008
%F A007814 a(n) = floor(A002487(n - 1) / A002487(n)). [From Reikku Kulon (reikku(AT)gmail.com), 
               Oct 05 2008]
%F A007814 Sum(k=1,n, (-1)^A000120(n-k)*a(k))=(-1)^(A000120(n)-1)*(A000120(n)-A000035(n)). 
               [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Mar 17 2009]
%F A007814 a(A001147(n)+A057077(n-1))=a(2n) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), 
               Mar 21 2009]
%F A007814 For n>=1, a(A004760(n+1))=a(n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), 
               Apr 15 2009]
%F A007814 2^(a(n))=A006519(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Apr 22 2009]
%F A007814 a(C(n,k))=A000120(k)+A000120(n-k)-A000120(n). [From Vladimir Shevelev 
               (shevelev(AT)bgu.ac.il), Jul 19 2009]
%F A007814 a(n!)=n-A000120(n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), 
               Jul 20 2009]
%e A007814 2^3 divides 24, so a(24)=3.
%e A007814 Contribution from Omar E. Pol (info(AT)polprimos.com), Jun 12 2009: (Start)
%e A007814 Triangle begins:
%e A007814 0;
%e A007814 1,0;
%e A007814 2,0,1,0;
%e A007814 3,0,1,0,2,0,1,0;
%e A007814 4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0;
%e A007814 5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0;
%e A007814 6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,
               0,2,...
%e A007814 (End)
%p A007814 ord:=proc(n) local i,j; if n=0 then RETURN(0); fi; i:=0; j:=n; while 
               j mod 2 <> 1 do i:=i+1; j:=j/2; od: i; end;
%t A007814 Table[IntegerExponent[n, 2], {n, 64}] (From Eric Weisstein)
%t A007814 p=2; Array[ If[ Mod[ #, p ]==0, Select[ FactorInteger[ # ], Function[ 
               q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 96 ]
%t A007814 DigitCount[BitXor[x, x - 1], 2, 1] - 1; a different version based on 
               the same concept: Floor[Log[2, BitXor[x, x - 1]]] (from Jaume Simon 
               Gispert (jaume(AT)nuem.com), Aug 29 2004)
%t A007814 Nest[Join[ # , ReplacePart[ # , Length[ # ] -> Last[ # ] + 1]] &, {0, 
               1}, 5] (from N. J. Gunther, May 23 2009)
%o A007814 (PARI) a(n)=valuation(n,2)
%Y A007814 A053398(1, n)
%Y A007814 a(n) = A001511[n]-1, column/row 1 of table A050602. Cf. A006519, A001511.
%Y A007814 a(2n)=A050603(2n).
%Y A007814 First differences of A011371. Bisection of A050605 and |A088705|.
%Y A007814 Cf. A122840, A122841, A007949, A112765.
%Y A007814 See A050603 and A136480 for a(n)+a(n+1).
%Y A007814 This is Guy Steele's sequence GS(1, 4) (see A135416).
%Y A007814 Cf. A002487 [From Reikku Kulon (reikku(AT)gmail.com), Oct 05 2008]
%Y A007814 A063787 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009]
%Y A007814 Cf. A000079. [From Omar E. Pol (info(AT)polprimos.com), Jun 12 2009]
%Y A007814 Sequence in context: A093057 A065334 A162590 this_sequence A083280 A060689 
               A053119
%Y A007814 Adjacent sequences: A007811 A007812 A007813 this_sequence A007815 A007816 
               A007817
%K A007814 nonn,nice,easy,new
%O A007814 1,4
%A A007814 John Tromp (tromp(AT)math.uwaterloo.ca)