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A001008
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Wolstenholme numbers: numerator of harmonic number H(n)=Sum_{i=1..n} 1/i.
(Formerly M2885 N1157)
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200
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1, 3, 11, 25, 137, 49, 363, 761, 7129, 7381, 83711, 86021, 1145993, 1171733, 1195757, 2436559, 42142223, 14274301, 275295799, 55835135, 18858053, 19093197, 444316699, 1347822955, 34052522467, 34395742267, 312536252003, 315404588903, 9227046511387
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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.
By Wolstenholme's theorem, p^2 divides a(p-1) for all primes p > 3.
From Alexander Adamchuk, Dec 11 2006: (Start)
p divides a(p^2-1) for all primes p>3.
p divides a((p-1)/2) for primes p in A001220.
p divides a((p+1)/2) or a((p-3)/2) for primes p in A125854.
a(n) is prime for n in A056903. Corresponding primes are given by A067657. (End)
a(n+1)= numerator of amazing polynomial A[1,n](1) where amazing polynomial A[genus 1,level n](m) is defined as Sum[m^(n - d)/d] d=1..n-1 Mathematica procedure generating A[1,n](m)is: m =.; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, k], {r, 1, 20}]; aa [Artur Jasinski, Oct 16 2008]
Better solutions to the card stacking problem have been found by M. Paterson and U. Zwick (see link). [Hugo Pfoertner, Jan 01 2012]
a(n) = A213999(n,n-1). - Reinhard Zumkeller, Jul 03 2012
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REFERENCES
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Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 259.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 347.
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences 3 (2000).
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
D. H. Bailey, J. M. Borwein, R. Girgensohn, Experimental evaluation of Euler sums, Exper. Math. 3 (1) (1994) 17, evaluate constants sum_{k>=1} H_k^m/(k+1)^n.
R. M. Dickau, Harmonic numbers and the book-stacking problem
Fredrik Johansson, How (not) to compute harmonic numbers. Feb 21 2009.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), Arxiv preprint arXiv:1111.3057, 2011
Hisanori Mishima, Factorizations of Wolstenholme numbers, n=1..100,
n=101..200, n=201..300.
M. Paterson et al, Maximum Overhang
N. J. A. Sloane, Illustration of initial terms
Eric Weisstein's World of Mathematics, Book Stacking Problem, Harmonic Number, Wolstenholme's Theorem, Harmonic Mean
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FORMULA
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H(n) ~ log n + gamma + O(1/n) [see for example Hardy and Wright, Th. 422.]
log n + gamma - 1/n < H(n) < log n + gamma + 1/n [follows easily from Hardy and Wright, Th. 422] - David Applegate and N. J. A. Sloane, Oct 14 2008
G.f. for H(n) : log(1-x)/(x-1) - Benoit Cloitre, Jun 15 2003
H(n) = Sqrt[Sum[Sum[1/(i*j), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk, Oct 24 2004
a(n) = Numerator[EulerGamma/n + PolyGamma[0, 1 + n]/n] [Artur Jasinski, Nov 02 2008]
H(n) = 3/2 + 2*sum(binomial(k+2,2)/(n-2-k),k=0..n-3)/((n-1)*n), n>1.[Gary Detlefs, Aug 02 2011]
H(n) = (-1)^(n-1)*((n+1)*n*sum(k=0..n-1, (k!*stirling2(n-1,k) * stirling1(n+k+1,n+1))/(n+k+1)!)). [Vladimir Kruchinin, Feb 05 2013]
H(n) = n*sum((-1)^k*binomial(n-1,k)/(k+1)^2,k=0..n-1). (Wenchang Chu). [Gary Detlefs, Apr 13 2013]
H(n) = 1/2*sum((-1)^(k-1)*binomial(n,k)*binomial(n+k,k)/k,k=1..n). (H. W. Gould) [Gary Detlefs, Apr 13 2013]
E.g.f. for H(n) = a(n)/A002805(n): (gamma + ln(x) - Ei(-x)) * exp(x), where gamma is the Euler-Mascheroni constant, and Ei(x) is the exponential integral [Vladimir Reshetnikov, Apr 24 2013]
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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,... ].
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MAPLE
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ZL:=n->sum(1/i, i=1..n): a:=n->floor(numer(ZL(n))): seq(a(n), n=1..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2007
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MATHEMATICA
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Table[Numerator[HarmonicNumber[n]], {n, 30}]
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PROG
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(PARI) A001008(n) = numerator(sum(i=1, n, 1/i)) \\ Michael B. Porter, Dec 08 2009
(Haskell)
import Data.Ratio ((%), numerator)
a001008 = numerator . sum . map (1 %) . enumFromTo 1
a001008_list = map numerator $ 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 A001008(n) : H = harmonic(1, n+1); return numerator(H[0]/H[1])
[A001008(n) for n in (1..29)] # Peter Luschny, Sep 01 2012
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CROSSREFS
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Cf. A002805 (denominators), A007406, A007408, A007410, A075135, A001220, A125854, A121999, A014566, A056903, A067657, A177427, A177690.
Cf. A145609-A145640. [From Artur Jasinski, Oct 16 2008]
Sequence in context: A060746 A111935 A175441 * A096617 A025529 A124078
Adjacent sequences: A001005 A001006 A001007 * A001009 A001010 A001011
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KEYWORD
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nonn,easy,frac,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Edited by Max Alekseyev, Oct 21 2011
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STATUS
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approved
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