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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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360
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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, 2329089562800, 72201776446800
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OFFSET
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1,2
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COMMENTS
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H(n)/2 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)/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 answer as a fraction of the time unit.
In general, if the k-th pipe needs d(k) days to fill the empty cistern then all pipes together need 1/Sum_{k=1..n} 1/d(k) = 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. On p. 97 of this reference one finds remarks on the history of such problems (called in German 'Brunnenaufgabe'). (End)
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)
All terms > 1 are even while corresponding numerators (A001008) are all odd (proof in Pólya and Szegő). - Bernard Schott, Dec 24 2021
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REFERENCES
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Chiu Chang Suan Shu, Neun Bücher arithmetischer Technik, translated 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.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, volume II, Springer, reprint of the 1976 edition, 1998, problem 251, p. 154.
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).
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LINKS
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FORMULA
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a(n) = Denominator(Sum_{k=1..n} (2*k-1)/k). - Gary Detlefs, Jul 18 2011
a(n) = the (reduced) denominator of the continued fraction 1/(1 - 1^2/(3 - 2^2/(5 - 3^2/(7 - ... - (n-1)^2/(2*n-1))))). - Peter Bala, Feb 18 2024
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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); # 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 *)
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PROG
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(Haskell)
import Data.Ratio ((%), denominator)
a002805 = denominator . sum . map (1 %) . enumFromTo 1
a002805_list = map denominator $ scanl1 (+) $ map (1 %) [1..]
(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])
(Magma) [Denominator(HarmonicNumber(n)): n in [1..40]]; // Vincenzo Librandi, Apr 16 2015
(GAP) List([1..30], n->DenominatorRat(Sum([1..n], i->1/i))); # Muniru A Asiru, Dec 20 2018
(Python)
from fractions import Fraction
def a(n): return sum(Fraction(1, i) for i in range(1, n+1)).denominator
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CROSSREFS
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KEYWORD
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nonn,easy,frac,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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