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A002805
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Denominators of harmonic numbers H(n)=Sum_{i=1..n} 1/i.
(Formerly M1589 N0619)
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136
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1, 2, 6, 12, 60, 20, 140, 280, 2520, 2520, 27720, 27720, 360360, 360360, 360360, 720720, 12252240, 4084080, 77597520, 15519504, 5173168, 5173168, 118982864, 356948592, 8923714800, 8923714800, 80313433200, 80313433200, 2329089562800
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OFFSET
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1,2
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COMMENTS
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H(n) is the maximal distance that a stack of n cards can project beyond the edge of a table without toppling.
If n is not in {1,2,6} then a(n) has at least one prime factor other then 2 or 5 . E.g. a(5)=60 has a prime factor 3 and a(7)=140 has a prime factor 7. This implies that every H(n)=A001008(n)/A002805(n), n not from {1,2,6}, has an infinite decimal representation. For a proof see the J. Havil reference. - W. Lang, Jun 29 2007
a(n) = A213999(n,n-1). - Reinhard Zumkeller, Jul 03 2012
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 259.
J. Havil, Gamma, (in German), Springer, 2007, p. 35-6; Gamma: Exploring Euler's Constant, Princeton Univ. Press, 2003.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 615.
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
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T. D. Noe, Table of n, a(n) for n=1..200
R. M. Dickau, Harmonic numbers and the book stacking problem
Fredrik Johansson, How (not) to compute harmonic numbers. Feb 21 2009.
N. J. A. Sloane, Illustration of initial terms
Eric Weisstein's World of Mathematics, Book Stacking Problem
Eric Weisstein's World of Mathematics, Harmonic Number
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FORMULA
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a(n) = Denominator(k=1..n, sum((2*k-1)/k) ). [From Gary Detlefs, Jul 18 2011]
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EXAMPLE
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H(n) = [ 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520,... ] = A001008/A002805.
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MAPLE
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seq(denom(sum((2*k-1)/k, k=1..n), n=1..30) [From Gary Detlefs, Jul 18 2011]
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MATHEMATICA
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Denominator[ Drop[ FoldList[ #1 + 1/#2 &, 0, Range[ 30 ] ], 1 ] ] - Harvey P. Dale Feb 09 2000
Table[Denominator[HarmonicNumber[n]], {n, 1, 40}] - Stefan Steinerberger, Apr 20 2006
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PROG
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(PARI) a(n)=denominator(sum(k=2, n, 1/k)) \\ Charles R Greathouse IV, Feb 11 2011
(Haskell)
import Data.Ratio ((%), denominator)
a002805 = denominator . sum . map (1 %) . enumFromTo 1
a002805_list = map denominator $ scanl1 (+) $ map (1 %) [1..]
-- Reinhard Zumkeller, Jul 03 2012
(Sage)
def harmonic(a, b): # See the F. Johansson link.
if b - a == 1 : return 1, a
m = (a+b)//2
p, q = harmonic(a, m)
r, s = harmonic(m, b)
return p*s+q*r, q*s
def A002805(n) : H = harmonic(1, n+1); return denominator(H[0]/H[1])
[A002805(n) for n in (1..29)] # Peter Luschny, Sep 01 2012
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CROSSREFS
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Cf. A001008 (numerators), A075135.
Sequence in context: A119862 A111936 A203811 * A117481 A083268 A085911
Adjacent sequences: A002802 A002803 A002804 * A002806 A002807 A002808
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KEYWORD
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nonn,easy,frac,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Stefan Steinerberger, Apr 20 2006
Definition edited by Daniel Forgues, May 19 2010
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STATUS
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approved
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