Search: id:A006519
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%I A006519 M0162
%S A006519 1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,
%T A006519 1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,64,1,2,1,4,
%U A006519 1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,1,4,1,2
%N A006519 Highest power of 2 dividing n.
%C A006519 Least positive k such that m^k+1 divides m^n+1(with fixed base m). - 
               Vladimir Baltic (baltic(AT)matf.bg.ac.yu), Mar 25 2002
%C A006519 To construct the sequence: start with 1, concatenate 1,1 and double last 
               term gives 1,2. Concatenate those 2 terms, 1,2,1,2 and double last 
               term 1,2,1,2 ->1,2,1,4. Concatenate those 4 terms: 1,2,1,4,1,2,1,
               4 and double last term -> 1,2,1,4,1,2,1,8 etc. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Dec 17 2002
%C A006519 a(n)=GCD(seq(binomial(2*n,2*m+1)/2,m=0..n-1)) (odd numbered entries of 
               even numbered rows of Pascal's triangle A007318 divided by 2), where 
               GCD denotes the greatest common divisor of a set of numbers. Due 
               to the symmetry of the rows it suffices to consider m=0..floor((n-1)/
               2). Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Jan 23 2004
%C A006519 Equals the continued fraction expansion of a constant x (cf. A100338) 
               such that the continued fraction expansion of 2*x interleaves this 
               sequence with 2's: contfrac(2*x) = [2; 1,2, 2,2, 1,2, 4,2, 1,2, 2,
               2, 1,2, 8,2,...].
%C A006519 Simon Plouffe (simon.plouffe(AT)gmail.com) observes that this sequence 
               and A003484 (Radon function) are very similar, the difference being 
               all zeros except for every 16-th term (see A101119 for nonzero differences). 
               Dec 02, 2004.
%C A006519 Comment from Jim Caprioli, Feb 04 2005: This sequence arises when calculating 
               the next odd number in a Collatz sequence: Next(x) = (3 * x + 1) 
               / A006519, or simply (3 x + 1) / BitAnd(3x+1,-3x-1).
%C A006519 a(n) = n iff n = 2^k. This sequence can be obtained by taking a(2^n) 
               = 2^n inplace of a(2^n) = n and using the same sequence building 
               approach as in A001511. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Jul 08 2005
%C A006519 Also smallest m such that m + n - 1 = m XOR (n - 1); A086799(n)=a(n)+n-1. 
               - Reinhard Zumkeller, Feb 02 2007
%C A006519 Number of 1's between successive 0's in A159689. [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Apr 22 2009]
%C A006519 Least number k such that all coefficients of k*E(n,x), the n-th Euler 
               polynomial, are integers (Cf. A144845). [From Peter Luschny (peter(AT)luschny.de), 
               Nov 13 2009]
%D A006519 R. Brown and J. L. Merzel, The number of Ducci sequences with a given 
               period, Fib. Quart., 45 (2007), 115-121.
%D A006519 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006519 T. D. Noe, <a href="b006519.txt">Table of n, a(n) for n=1..10000</a>
%H A006519 Beeler, M., Gosper, R. W. and Schroeppel, R., <a href="http://www.inwap.com/
               pdp10/hbaker/hakmem/hacks.html#item175">Item 175 in Beeler, M., Gosper, 
               R. W. and Schroeppel, R. HAKMEM. MIT AI Memo 239, Feb. 29, 1972</
               a>
%H A006519 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A006519 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%H A006519 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer 
               generating functions. I. Elementary sequences</a>
%H A006519 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EvenPart.html">Link to a section of The World of Mathematics.</a>
%F A006519 a(n) = n AND -n (where "AND" is bitwise) - Marc LeBrun (mlb(AT)well.com), 
               Sep 25 2000
%F A006519 Also: a(n)=gcd[2^n, n]. - Labos E. (labos(AT)ana.sote.hu), Apr 22 2003
%F A006519 Multiplicative with a(p^e) = p^e if p = 2; 1 if p > 2. - David W. Wilson 
               (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A006519 G.f.: sum(k>=0, 2^k*x^2^k/(1-x^2^(k+1))). - Ralf Stephan, May 06 2003
%F A006519 Dirichlet g.f.: zeta(s)*(2^s-1)/(2^s-2). - Ralf Stephan, Jun 17 2007
%F A006519 a(n) = 2^floor(A002487(n - 1) / A002487(n)). [From Reikku Kulon (reikku(AT)gmail.com), 
               Oct 05 2008]
%e A006519 2^3 divides 24, but 2^4 does not divide 24, so a(24)=8.
%p A006519 with(numtheory): for n from 1 to 200 do if n mod 2 = 1 then printf(`%d,
               `,1) else printf(`%d,`,2^ifactors(n)[2][1][2]) fi; od:
%t A006519 f[n_] := Block[{k = 0}, While[Mod[n, 2^k] == 0, k++ ]; 2^(k - 1)]; Table[ 
               f[n], {n, 102}] (from Robert G. Wilson v Nov 17 2004)
%o A006519 (PARI) a(n)=2^valuation(n,2)
%Y A006519 Partial sums are in A006520, second partial sums in A022560.
%Y A006519 Cf. A007814, A100338, A003484, A101119.
%Y A006519 This is Guy Steele's sequence GS(5, 2) (see A135416).
%Y A006519 Cf. A002487 [From Reikku Kulon (reikku(AT)gmail.com), Oct 05 2008]
%Y A006519 Sequence in context: A084236 A068057 A003484 this_sequence A055975 A118827 
               A118830
%Y A006519 Adjacent sequences: A006516 A006517 A006518 this_sequence A006520 A006521 
               A006522
%K A006519 nonn,easy,nice,mult,new
%O A006519 1,2
%A A006519 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A006519 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 20 2000

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