%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]
%D A000265 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000265 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000265 Problem H-81, Fib. Quart., 6 (1968), 52.
%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.
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