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A002805 Denominators of harmonic numbers H(n)=Sum_{i=1..n} 1/i.
(Formerly M1589 N0619)
175
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

From Wolfdieter Lang, Apr 16 2015: (Start)

a(n)/A001008(n) = 1/H(n) is the solution of the following version of the classical cistern and pipes problem. A cistern is connected to n different pipes of water. For the k-th pipe it takes k time units (say, days) to fill the empty cistern, for k = 1, 2, ..., n. How long does it take for the n pipes together to fill the empty cistern? 1/H(n) gives the fraction of a the time unit as answer.

In general, if the k-th pipe needs d(k) days to fill the empty cistern then all pipes together need 1/sum(1/d(k),k=1..n) =  HM(d(1), ... ,d(n))/n days, where HM denotes the harmonic mean HM. For the described problem, HM(1, 2, ..., n)/n = A102928(n)/(n*A175441(n)) = 1/H(n).

For a classical cistern and pipes problem see, e.g., the Hunger-Vogel reference (in Greek and German) given in A256101, problem 27, p. 29, where n = 3, and d(1), d(2) and d(3) are 6, 4 and 3 days. 0n p. 97 of this reference one finds remarks on the history of such problems (called in German 'Brunnenaufgabe').(End)

From Wolfdieter Lang, Apr 17 2015: (Start)

An example of the above mentioned cistern and pipes problems appears in Chiu Chang Suan Shu (nine books on arithmetic) in book VI, problem 26. The numbers are there 1/2, 1, 5/2, 3 and 5 (days) and the result is 15/75 (day). See the reference (in German) on p. 68.

A historical account on such cistern problems is found in the Johannes Tropfke reference, given in A256101, section 4.2.1.2 Zisternenprobleme (Leistungsprobleme), pp. 578-579.

In Fibonacci's Liber Abaci such problems appear on p. 281 and p. 284 of L. E. Sigler's translation. (End)

REFERENCES

Chiu Chang Suan Shu, Neun Bücher arithmetischer Technik, trnaslated and commented by Kurt Vogel, Ostwalds Klassiker der exakten Wissenschaften, Band 4, Friedr. Vieweg & Sohn, Braunschweig, 1968, p. 68.

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.

L. E. Sigler, Fibonacci's Liber Abaci, Springer, 2003, pp. 281, 284.

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

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

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

(MAGMA) [Denominator(HarmonicNumber(n)): n in [1..40]]; // Vincenzo Librandi, Apr 16 2015

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 July 3 11:36 EDT 2015. Contains 259161 sequences.