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A001008 Wolstenholme numbers: numerator of harmonic number H(n)=Sum_{i=1..n} 1/i.
(Formerly M2885 N1157)
220
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 (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.

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 polynomial A[1,n](1) where 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

From Paul Curtz, Nov 30 2013: (Start)

b(n)=0 followed by H(n)(=1, 3/2,...) = 0, 1, 3/2, 11/6,... .

Akiyama-Tanigawa algorithm applied to b(n):

0,   1, 3/2, 11/6, 25/12, 137/60,...

-1, -1,  -1,   -1,    -1,     -1,... =-A000012.

0,   0,   0,    0,     0,      0,... =A000004.

The first column is -A063524.

c(n)=0 followed by b(n). The difference table of c(n) is:

0,         0,    1,   3/2, 11/6, 25/12,...

0,         1,  1/2,   1/3,  1/4,...     =A211666?/A028310,

1,      -1/2, -1/6, -1/12,...

-3/2,    1/3, 1/12,...

11/6,   -1/4,...

-25/12,... .

c(n) is an autosequence of second kind: its inverse binomial transform is the signed sequence with the main diagonal double of the first upper diagonal. (End)

REFERENCES

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.

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

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.

Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.

R. M. Dickau, Harmonic numbers and the book-stacking problem

Fredrik Johansson, How (not) to compute harmonic numbers. Feb 21 2009.

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

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

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

Eric Weisstein's World of Mathematics, Book Stacking Problem, Wolstenholme's Theorem, Harmonic Mean

FORMULA

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]

H(n) = residue((psi(-s)+gamma)^2/2, {s, n}) where psi is the digamma function and gamma the Euler-Mascheroni constant. - Jean-François Alcover, Feb 19 2014

EXAMPLE

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

MAPLE

ZL:=n->sum(1/i, i=1..n): a:=n->floor(numer(ZL(n))): seq(a(n), n=1..26); - Zerinvary Lajos, Mar 28 2007

MATHEMATICA

Table[Numerator[HarmonicNumber[n]], {n, 30}]

PROG

(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

CROSSREFS

Cf. A002805 (denominators), A007406, A007408, A007410, A075135, A001220, A125854, A121999, A014566, A056903, A067657, A177427, A177690.

Cf. A145609-A145640. [From Artur Jasinski, Oct 16 2008]

Cf. A003506. [From Paul Curtz, Nov 30 2013]

Sequence in context: A060746 A111935 A175441 * A231606 A096617 A025529

Adjacent sequences:  A001005 A001006 A001007 * A001009 A001010 A001011

KEYWORD

nonn,easy,frac,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Edited by Max Alekseyev, Oct 21 2011

STATUS

approved

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Last modified April 24 02:19 EDT 2014. Contains 240947 sequences.