%I A000265 M2222 N0881
%S A000265 1,1,3,1,5,3,7,1,9,5,11,3,13,7,15,1,17,9,19,5,21,11,23,3,25,13,27,7,29,
%T A000265 15,31,1,33,17,35,9,37,19,39,5,41,21,43,11,45,23,47,3,49,25,51,13,53,
%U A000265 27,55,7,57,29,59,15,61,31,63,1,65,33,67,17,69,35,71,9,73,37,75,19,77
%N A000265 Remove 2's from n; or largest odd divisor of n; or odd part of n.
%C A000265 When n>0 is written as k*2^j with k odd then k=A000265(n) and j=A007814(n), 
               so: when n is written as k*2^j-1 with k odd then k=A000265(n+1) and 
               j=A007814(n+1), when n>1 is written as k*2^j+1 with k odd then k=A000265(n-1) 
               and j=A007814(n-1)
%C A000265 Also denominator of 2^n/n (numerator is A075101(n)). - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Sep 01 2002
%C A000265 Slope of line connecting (o,a(o)) where o=(2^k)(n-1)+1 is 2^k and (by 
               design) starts at (1,1) - Josh Locker (joshlocker(AT)macfora.com), 
               Apr 17 2004
%C A000265 Numerator of n/2^(n-1). - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Feb 11 2005
%C A000265 Comment from Marco Matosic (marcomatosic(AT)hotmail.com), Jun 29 2005:
%C A000265 "The sequence can be arranged in a table:
%C A000265 ...................................1
%C A000265 ................................1..3..1
%C A000265 ............................1...5..3..7...1
%C A000265 ....................1...9...5..11..3..13..7...15..1
%C A000265 ......1..17..9..19..5..21..11..23..3..25..13..27..7..29..15..31..1
%C A000265 Every new row is the previous row interspaced with the continuation of 
               the odd numbers.
%C A000265 Except for the ones; the terms (t) in each column are t+t+/-s = t_+1. 
               Starting from the center column of threes and working to the left 
               the values of s are given by A000265 and working to the right by 
               A000265."
%C A000265 (a(k),a(2k),a(3k),...)=a(k)*(a(1),a(2),a(3),...) In general, a[n*m]=a[n]*a[m] 
               - Josh Locker (jlocker(AT)mail.rochester.edu), Oct 04 2005
%C A000265 This is a fractal sequence. The odd-numbered elements give the odd natural 
               numbers. If these elements are removed, the original sequence is 
               recovered. - Kerry Mitchell (lkmitch(AT)gmail.com), Dec 07 2005
%C A000265 2k+1 is the k-th and largest of the subsequence of k terms separating 
               two successive equal entries in a(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Dec 30 2005
%C A000265 It's not difficult to show that the sum of the first 2^n terms is (4^n 
               + 2)/3. - Nick Hobson, Jan 14 2005
%C A000265 a(A132739(n)) = A132739(a(n)) = A132740(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 27 2007
%C A000265 Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), May 27 2009: 
               (Start)
%C A000265 In the table, for each row,
%C A000265 (sum of terms between 3 and 1) - (sum of terms between 1 and 3) = A020988. 
               (End)
%C A000265 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%C A000265 This sequence appears in the analysis of the 'look-a-likes' of the numerator 
               and denominator of the Taylor series for tan(x), i.e. A160469(n) 
               and A156769(n).
%C A000265 (End)
%C A000265 a(n)=n/gcd(2^n,n). (This also shows that the true offset is 0 and a(0)=0.) 
               [From Peter Luschny (peter(AT)luschny.de), Nov 14 2009]
%C A000265 Consider 1) 0-th, 2) 2nd, 3) 4-th, 4) 6-th, , n+1) 2n-th, numerators 
               of Rydberg-Ritz spectra of atomic hydrogen: A000012,A061037(Balmer),
               A061041(Brackett),A061045(Humphreys),A061049,followers calculated 
               by Richard Mathar,April 29;from 2nd ( 2)) we take itself (monosection),
               bisection (for 4th) hence 3b) ,trisection hence 4t),quadrisection,
               quintisection or pentasection,hexasection,heptasection,octosection, 
               .Then we take first term of 1), first two of 2), first three of 3b), 
               first four of 4t), .. . Hence triangle 1; 0,5; 0,5,3; 0,5,1,7; 0,
               5,3,21,1; 0,1,3,21,2,9; 0,5,1,7,1,5,5; 0,5,3,3,2,45,15,11; 0,5,3,
               21,1,45,15,77,3; Last term of each row gives 1,5,3,7,1,9,5,11,3,=A000265(n+3).For 
               0-th spectrum see A152977 and A165795 (comment must be corrected). 
               a(n) taken by A000079=2^n terms gives sum 1,4,16,64,=A000302. [From 
               Paul Curtz (bpcrtz(AT)free.fr), Nov 27 2009]
%D A000265 Problem H-81, Fib. Quart., 6 (1968), 52.
%D A000265 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000265 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000265 T. D. Noe, <a href="b000265.txt">Table of n, a(n) for n=1..10000</a>
%H A000265 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A000265 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%H A000265 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               OddPart.html">Link to a section of The World of Mathematics.</a>
%H A000265 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TrigonometryAngles.html">Trigonometry Angles</a>
%H A000265 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SphereLinePicking.html">Sphere Line Picking</a>
%H A000265 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               OddPart.html">Odd Part</a>
%F A000265 a(n) = if n is odd then n else a(n/2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Sep 01 2002
%F A000265 a(n) = n/A006519(n) = 2*A025480(n-1)+1
%F A000265 Multiplicative with a(p^e) = 1 if p = 2, p^e if p > 2. - David W. Wilson 
               (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000265 a(n) = Sum_{d divides n and d is odd} phi(d). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Dec 04 2002
%F A000265 G.f.: -1/(1-x) + sum(k>=0, 2x^2^k/(1-2x^2^(k+1)+x^2^(k+2))). - Ralf Stephan 
               (ralf(AT)ark.in-berlin.de), Sep 05 2003
%F A000265 Dirichlet g.f.: zeta(s-1)*(2^s-2)/(2^s-1). - R. Stephan, Jun 18 2007
%F A000265 a(n)=sum{k=0..n, A127793(n,k)*floor((k+2)/2)} (conjecture). - Paul Barry 
               (pbarry(AT)wit.ie), Jan 29 2007
%F A000265 a(n) = 2*A003602(n) - 1. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jul 02 2009]
%p A000265 A000265:=proc(n) local t1,d; t1:=1; for d from 1 by 2 to n do if n mod 
               d = 0 then t1:=d; fi; od; t1; end;
%t A000265 Table[Times@@(#[[1]]^#[[2]]&/@Select[FactorInteger[i], #[[1]]!=2&]), 
               {i, 90}] (from Harvey Dale)
%t A000265 a[n_Integer /; n > 0] := n/2^IntegerExponent[n, 2] (Josh Locker)
%o A000265 (PARI) a(n)=if(n<1, 0, n/2^valuation(n, 2)) /* Michael Somos Aug 09 2006 
               */
%Y A000265 Cf. A111929, A111930, A111918, A111919, A111920, A111921, A111922, A111923.
%Y A000265 Cf. A038502, A065330, A135013.
%Y A000265 Sequence in context: A098985 A072963 A161955 this_sequence A106617 A040026 
               A106609
%Y A000265 Adjacent sequences: A000262 A000263 A000264 this_sequence A000266 A000267 
               A000268
%K A000265 mult,nonn,easy,nice,new
%O A000265 1,3
%A A000265 N. J. A. Sloane (njas(AT)research.att.com).
%E A000265 Additional comments from Henry Bottomley (se16(AT)btinternet.com), Mar 
               02 2000. More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 
               2000.