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A002805 Denominators of harmonic numbers H(n)=Sum_{i=1..n} 1/i.
(Formerly M1589 N0619)
165
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 than 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. - Wolfdieter Lang, Jun 29 2007

a(n) = A213999(n,n-1). - Reinhard Zumkeller, Jul 03 2012

REFERENCES

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.

A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.

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).

LINKS

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

J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number

Eric Weisstein's World of Mathematics, Book Stacking Problem

FORMULA

a(n) = Denominator(k=1..n, sum((2*k-1)/k) ). - Gary Detlefs, Jul 18 2011

EXAMPLE

H(n) = [ 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520,... ] = A001008/A002805.

MAPLE

seq(denom(sum((2*k-1)/k, k=1..n), n=1..30); # Gary Detlefs, Jul 18 2011

f:=n->denom(add(1/k, k=1..n)); # - N. J. A. Sloane, Nov 15 2013

MATHEMATICA

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*)

PROG

(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

CROSSREFS

Cf. A001008 (numerators), A075135.

Sequence in context: A111936 A232090 A203811 * A231693 A232112 A117481

Adjacent sequences:  A002802 A002803 A002804 * A002806 A002807 A002808

KEYWORD

nonn,easy,frac,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Stefan Steinerberger, Apr 20 2006

Definition edited by Daniel Forgues, May 19 2010

STATUS

approved

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Last modified September 21 11:48 EDT 2014. Contains 247025 sequences.