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A000265
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Remove 2's from n; or largest odd divisor of n; or odd part of n.
(Formerly M2222 N0881)
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117
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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, 15, 31, 1, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 1, 65, 33, 67, 17, 69, 35, 71, 9, 73, 37, 75, 19, 77
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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)
Also denominator of 2^n/n (numerator is A075101(n)). - Reinhard Zumkeller, Sep 01 2002
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
Numerator of n/2^(n-1). - Alexander Adamchuk, Feb 11 2005
Comment from Marco Matosic (marcomatosic(AT)hotmail.com), Jun 29 2005:
"The sequence can be arranged in a table:
...................................1
................................1..3..1
............................1...5..3..7...1
....................1...9...5..11..3..13..7...15..1
......1..17..9..19..5..21..11..23..3..25..13..27..7..29..15..31..1
Every new row is the previous row interspaced with the continuation of the odd numbers.
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."
(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
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
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
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
a(A132739(n)) = A132739(a(n)) = A132740(n). - Reinhard Zumkeller, Aug 27 2007
Contribution from Eric Desbiaux, May 27 2009: (Start)
In the table, for each row,
(sum of terms between 3 and 1) - (sum of terms between 1 and 3) = A020988. (End)
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009: (Start)
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).
(End)
a(n)=n/gcd(2^n,n). (This also shows that the true offset is 0 and a(0)=0.) [From Peter Luschny, Nov 14 2009]
It seems that in most cases A030101[x]=A000265[x] and that if A030101[x]<>A000265[x], the next time A030101[y]=A000265[x], A030101[x]=A000265[y]. also, it seems that if a pair of values exist at one index, they will exist at any index where one of them exist. it also seems like te greater of the pair always shows up on A000265 first. [From Dylan Hamilton (PhalarisBull(AT)gmail.com), Aug 04 2010]
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REFERENCES
| Problem H-81, Fib. Quart., 6 (1968), 52.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Daniel Forgues, Table of n, a(n) for n=1..100000
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Trigonometry Angles
Eric Weisstein's World of Mathematics, Sphere Line Picking
Eric Weisstein's World of Mathematics, Odd Part
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FORMULA
| a(n) = if n is odd then n, else a(n/2). - Reinhard Zumkeller, Sep 01 2002
a(n) = n/A006519(n) = 2*A025480(n-1)+1
Multiplicative with a(p^e) = 1 if p = 2, p^e if p > 2. - David W. Wilson, Aug 01, 2001.
a(n) = Sum_{d divides n and d is odd} phi(d). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 04 2002
G.f.: -1/(1-x) + sum(k>=0, 2x^2^k/(1-2x^2^(k+1)+x^2^(k+2))). - Ralf Stephan, Sep 05 2003
Dirichlet g.f.: zeta(s-1)*(2^s-2)/(2^s-1). - R. Stephan, Jun 18 2007
a(n)=sum{k=0..n, A127793(n,k)*floor((k+2)/2)} (conjecture). - Paul Barry, Jan 29 2007
a(n) = 2*A003602(n) - 1. [From Franklin T. Adams-Watters, Jul 02 2009]
a(-n) = -a(n). - Michael Somos, Sep 19 2011
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EXAMPLE
| x + x^2 + 3*x^3 + x^4 + 5*x^5 + 3*x^6 + 7*x^7 + x^8 + 9*x^9 + 5*x^10 + 11*x^11 + ...
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MAPLE
| 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;
A000265 := n -> n/2^padic[ordp](n, 2): [Peter Luschny, Nov 26 2010]
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MATHEMATICA
| Table[Times@@(#[[1]]^#[[2]]&/@Select[FactorInteger[i], #[[1]]!=2&]), {i, 90}] (* from Harvey Dale *)
a[n_Integer /; n > 0] := n/2^IntegerExponent[n, 2]; Array[a, 77] (* Josh Locker *)
f[n_] := n/2^IntegerExponent[n, 2]; Array[f, 77] (* From Robert G. Wilson v, Nov 07 2010 *)
f[n_] := Denominator[2^n/n]; Array[f, 100] (* From Vladimir Joseph Stephan Orlovsky, Feb 16 2011 *)
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PROG
| (PARI) {a(n) = if( n==0, 0, n / 2^valuation(n, 2))} /* Michael Somos, Aug 09 2006 */
(Haskell)
a000265 n | r > 0 = n
| otherwise = a000265 n' where (n', r) = divMod n 2
-- Reinhard Zumkeller, Oct 14 2010, Apr 08 2011 (improved version)
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CROSSREFS
| Cf. A111929, A111930, A111918, A111919, A111920, A111921, A111922, A111923.
Cf. A038502, A065330, A135013.
Sequence in context: A098985 A072963 A161955 * A106617 A040026 A106609
Adjacent sequences: A000262 A000263 A000264 * A000266 A000267 A000268
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KEYWORD
| mult,nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| 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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