A007409 -id:A007409 - cp's OEIS frontend

A238167 Decimal expansion of sum_(n>=1) H(n,3)/n^5 where H(n,3) = A007408(n)/A007409(n) is the n-th harmonic number of order 3.

1, 0, 4, 6, 9, 2, 4, 4, 0, 1, 7, 2, 4, 6, 7, 6, 0, 8, 2, 3, 4, 5, 7, 2, 3, 0, 1, 4, 2, 2, 2, 7, 9, 2, 3, 2, 9, 6, 1, 9, 5, 9, 8, 4, 0, 2, 2, 6, 4, 1, 4, 7, 7, 1, 4, 7, 4, 8, 3, 3, 2, 5, 0, 9, 5, 0, 5, 1, 8, 3, 8, 4, 4, 2, 2, 8, 2, 0, 1, 1, 1, 9, 0, 0, 1, 7, 8, 1, 8, 5, 1, 8, 6, 0, 3, 0, 7, 7, 9, 7

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.046924401724676082345723014222792329619598402264...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 16.

Cf. A007408, A007409, A152648, A152649, A152651, A238166, A238168, A238169.

RealDigits[5*Zeta[2]*Zeta[5] +2*Zeta[3]*Zeta[4] -10*Zeta[7],10,100][[1]]

5*zeta(2)*zeta(5) + 2*zeta(3)*zeta(4) - 10*zeta(7) \\ G. C. Greubel, Dec 30 2017

Equals 5*zeta(2)*zeta(5) + 2*zeta(3)*zeta(4) - 10*zeta(7).

A244665 Decimal expansion of sum_(n>=1) (H(n,3)/n^3) where H(n,3) = A007408(n)/A007409(n) is the n-th harmonic number of order 3.

Original entry on oeis.org

1, 2, 3, 1, 1, 4, 1, 9, 3, 0, 2, 0, 9, 0, 4, 1, 6, 8, 6, 8, 1, 4, 1, 0, 1, 5, 0, 4, 2, 9, 8, 9, 5, 4, 1, 7, 7, 5, 4, 2, 7, 7, 6, 4, 4, 7, 8, 9, 8, 3, 7, 1, 1, 1, 7, 9, 8, 6, 9, 2, 1, 4, 1, 2, 9, 5, 1, 4, 5, 8, 0, 1, 9, 5, 1, 6, 6, 5, 5, 9, 9, 9, 9, 2, 4, 4, 8, 3, 5, 3, 8, 2, 2, 8, 5, 2, 6, 3, 2, 5, 5, 9, 5

Offset: 1

Jean-François Alcover, Jul 04 2014

			1.2311419302090416868141015042989541775427764478983711179869214129514580195...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 23.

Cf. A002117, A007408, A007409, A244664.

RealDigits[1/2*Zeta[3]^2 + 1/2*Zeta[6], 10, 103] // First

zeta(3)^2/2 + Pi^6/1890 \\ Michel Marcus, Jul 04 2014

zeta(3)^2/2 + Pi^6/1890.

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

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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) = 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 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)
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)
All terms > 1 are even while corresponding numerators (A001008) are all odd (proof in Pólya and Szegő). - Bernard Schott, Dec 24 2021

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

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.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 258-261.
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).

Kenny Lau, Table of n, a(n) for n = 1..2308 [First 200 terms computed by T. D. Noe]
R. M. Dickau, Harmonic numbers and the book stacking problem.
Haight, Frank A., and Robert B. Jones., "A probabilistic treatment of qualitative data with special reference to word association tests." Journal of Mathematical Psychology 11.3 (1974): 237-244. [Annotated scanned copy]
Frank Haight and N. J. A. Sloane, Correspondence, 1975.
Antal 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.
Peter Shiu, The denominators of harmonic numbers, arXiv:1607.02863 [math.NT], 2016.
N. J. A. Sloane, Illustration of initial terms.
Jonathan Sondow and Eric W. Weisstein, MathWorld: Harmonic Number.
Eric Weisstein's World of Mathematics, Book Stacking Problem.
Bing-Ling Wu, Yong-Gao Chen, On the denominators of harmonic numbers, arXiv:1711.00184 [math.NT], 2017.

Cf. A001008 (numerators), A075135, A025529, A203810, A203811, A203812.
Partial sums: A027612/A027611 = 1, 5/2, 13/3, 77/12, 87/10, 223/20,...
The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358, Sum 1/n^2: A007406/A007407, Sum 1/n^3: A007408/A007409.

List([1..30],n->DenominatorRat(Sum([1..n],i->1/i))); # Muniru A Asiru, Dec 20 2018

import Data.Ratio ((%), denominator)
a002805 = denominator . sum . map (1 %) . enumFromTo 1
a002805_list = map denominator $ scanl1 (+) $ map (1 %) [1..]
-- Reinhard Zumkeller, Jul 03 2012

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

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

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 *)
Denominator[Accumulate[1/Range[25]]] (* Alonso del Arte, Nov 21 2018 *)

a(n)=denominator(sum(k=2,n,1/k)) \\ Charles R Greathouse IV, Feb 11 2011

from fractions import Fraction
def a(n): return sum(Fraction(1, i) for i in range(1, n+1)).denominator
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, Dec 24 2021

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

a(n) = Denominator(Sum_{k=1..n} (2*k-1)/k). - Gary Detlefs, Jul 18 2011
a(n) = n! / gcd(Stirling1(n+1, 2), n!) = n! / gcd(A000254(n),n!). - Max Alekseyev, Mar 01 2018
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

Definition edited by Daniel Forgues, May 19 2010

A007408 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3.

Original entry on oeis.org

1, 9, 251, 2035, 256103, 28567, 9822481, 78708473, 19148110939, 19164113947, 25523438671457, 25535765062457, 56123375845866029, 56140429821090029, 56154295334575853, 449325761325072949, 2207911834254200646437, 245358578943756786493

Offset: 1

N. J. A. Sloane, Mira Bernstein

By Theorem 131 in Hardy and Wright, p^2 divides a(p - 1) for prime p > 5. - T. D. Noe, Sep 05 2002
p^3 divides a(p - 1) for prime p = 37. Primes p such that p divides a((p + 1)/2) are listed in A124787(n) = {3, 11, 17, 89}. - Alexander Adamchuk, Nov 07 2006
a(n)/A007409(n) is the partial sum towards zeta(3), where zeta(s) is the Riemann zeta function. - Alonso del Arte, Dec 30 2012
See the Wolfdieter Lang link under A103345 on Zeta(k, n) with the rationals for k=1..10, g.f.s and polygamma formulas. - Wolfdieter Lang, Dec 03 2013
Denominator of the harmonic mean of the first n cubes. - Colin Barker, Nov 13 2014

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1971, page 104.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..200
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
D. Y. Savio, E. A. Lamagna and S.-M. Liu, Summation of harmonic numbers, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.2.

Cf. A001008, A007406, A007409 (denom), A002117, A124787, A249950.

A007408:=n->numer(sum(1/k^3,k=1..n)); map(%,[$1..20]); # M. F. Hasler, Nov 10 2006

Table[Numerator[Sum[1/k^3, {k, n}]], {n, 10}] (* Alonso del Arte, Dec 30 2012 *)
Table[Denominator[HarmonicMean[Range[n]^3]],{n,20}] (* Harvey P. Dale, Aug 20 2017 *)
Accumulate[1/Range[20]^3]//Numerator (* Harvey P. Dale, Aug 28 2023 *)

a(n)=numerator(sum(k=1,n,1/k^3)) \\ Charles R Greathouse IV, Jul 19 2011

from fractions import Fraction
from itertools import accumulate, count, islice
def A007408gen(): yield from map(lambda x: x.numerator, accumulate(Fraction(1, k**3) for k in count(1)))
print(list(islice(A007408gen(), 20))) # Michael S. Branicky, Jun 26 2022

Sum_{k = 1 .. n} 1/k^3 = sqrt(sum_{j = 1 .. n} sum_{i = 1 .. n} 1/(i * j)^3). - Alexander Adamchuk, Oct 26 2004

A103345 Numerator of Sum_{k=1..n} 1/k^6 = Zeta(6,n).

Original entry on oeis.org

1, 65, 47449, 3037465, 47463376609, 47464376609, 5584183099672241, 357389058474664049, 260537105518334091721, 52107472322919827957, 92311616995117182948130877, 92311647383100199924330877, 445570781131605573859221176881493, 445570839299219762020391212081493

Offset: 1

Wolfdieter Lang, Feb 15 2005

For the rationals Zeta(k,n) for k = 1..10 and n = 1..20, see the W. Lang link.

a(n) gives the partial sum, Zeta(6,n), of Euler's (later Riemann's) Zeta(6). Zeta(k,n), k >= 2, is sometimes also called H(k,n) because for k = 1 these would be the harmonic numbers A001008/A002805. However, H(1,n) does not give partial sums of a convergent series.

			The first few fractions are 1, 65/64, 47449/46656, 3037465/2985984, 47463376609/46656000000, ... = A103345/A103346. - _Petros Hadjicostas_, May 10 2020

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, Rational Zeta(k,n) and more.

Cf. A013664, A291456. For the denominators, see A103346.

For k=1..5, see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052.

s=0; lst={}; Do[s+=n^1/n^7; AppendTo[lst,Numerator[s]],{n,3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 6] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(Sum_{k=1..n} 1/k^6) = numerator(A291456(n)/(n!)^6).
G.f. for rationals Zeta(6, n): polylogarithm(6, x)/(1-x).
Zeta(6, n) = (1/945)*Pi^6 - psi(5, n+1)/5!, see eq. 6.4.3 with m = 5, p. 260, of the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 03 2013

A103347 Numerators of Sum_{k=1..n} 1/k^7 = Zeta(7,n).

Original entry on oeis.org

1, 129, 282251, 36130315, 2822716691183, 940908897061, 774879868932307123, 99184670126682733619, 650750755630450535274259, 650750820166709327386387, 12681293156341501091194786541177, 12681293507322704937269896541177

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(7,n), of Euler's Zeta(7). Zeta(k,n) is also called H(k,n) because for k=1 these are the harmonic numbers H(n) A001008/A002805.

For the denominators see A103348 and for the rationals Zeta(7,n) see the W. Lang link under A103345.

Robert Israel, Table of n, a(n) for n = 1..336

For k=1..6 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346.

f:= n -> numer(Psi(6,n+1)/720 + Zeta(7)):
map(f, [$1..20]); # Robert Israel, Mar 28 2018

s=0;lst={};Do[s+=n^1/n^8;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 7] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(sum_{k=1..n} 1/k^7).

G.f. for rationals Zeta(7, n): polylogarithm(7, x)/(1-x).

A103349 Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).

Original entry on oeis.org

1, 257, 1686433, 431733409, 168646292872321, 168646392872321, 972213062238348973121, 248886558707571775009601, 1632944749460578249437992161, 1632944765723715465050248417

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(8,n) of Euler's Zeta(8). Zeta(k,n) is also called H(k,n) because for k=1 these are the harmonic numbers H(n) A001008/A002805.

For the denominators see A103350 and for the rationals Zeta(8,n) see the W. Lang link under A103345.

For k=1..7 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348.

s=0;lst={};Do[s+=n^1/n^9;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 8] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)
Accumulate[1/Range[10]^8]//Numerator (* Harvey P. Dale, Aug 11 2024 *)

a(n)=numerator(sum_{k=1..n} 1/k^8).

G.f. for rationals Zeta(8, n): polylogarithm(8, x)/(1-x).

A103351 Numerators of sum_{k=1..n} 1/k^9 = Zeta(9,n).

Original entry on oeis.org

1, 513, 10097891, 5170139875, 10097934603139727, 373997614931101, 15092153145114981831307, 7727182467755471289426059, 4106541588424891370931874221019, 4106541592523201949266162797531

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(9,n), of Euler's Zeta(9). Zeta(k,n) is also called H(k,n) because for k=1 these are the harmonic numbers H(n) A001008/A002805.

For the denominators see A103352 and for the rationals Zeta(9,n) see the W. Lang link under A103345.

For k=1..8 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348, A103349/A103350.

s=0;lst={};Do[s+=n^1/n^10;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 9] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(sum_{k=1..n} 1/k^9).

G.f. for rationals Zeta(9, n): polylogarithm(9, x)/(1-x).

A068589 Denominator(Sum_{i=1..n} 1/i^3)/denominator(Sum_{i=1..n} 1/i).

Original entry on oeis.org

1, 4, 36, 144, 3600, 1200, 58800, 235200, 6350400, 6350400, 768398400, 768398400, 129859329600, 129859329600, 129859329600, 519437318400, 150117385017600, 50039128339200, 18064125330451200, 18064125330451200, 54192375991353600

Offset: 1

Benoit Cloitre, Mar 27 2002

For n = 1 to n = 19, we have a(n) = A334580(n), but a(20) = 18064125330451200 <> 3612825066090240 = A334580(20). - Petros Hadjicostas, May 06 2020

Cf. A002805, A007409, A119682, A334580.

a := proc(n) local i: denom(add(1/i^3, i = 1 .. n))/denom(add(1/i, i = 1 .. n)): end proc:
seq(a(n), n=1..30); # Petros Hadjicostas, May 06 2020

a(n) = denominator(sum(k=1, n, 1/k^3)/sum(k=1, n, 1/k)); \\ Michel Marcus, May 07 2020

a(n) = A007409(n)/A002805(n).

A103716 Numerators of sum_{k=1..n} 1/k^10 =: Zeta(10,n).

Original entry on oeis.org

1, 1025, 60526249, 61978938025, 605263128567754849, 605263138567754849, 170971856382109814342232401, 175075181098169912564190119249, 10338014371627802833957102351534201, 413520574906423083987893722912609

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(10,n), of Euler's Zeta(10). Zeta(k,n) is also called H(k,n) because for k=1 these are the harmonic numbers H(n) = A001008/A002805.

For the denominators see A103717 and for the rationals Zeta(10,n) see the W. Lang link under A103345.

For k=1..9 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348, A103349/A103350, A103351/A103352.

s=0;lst={};Do[s+=n^1/n^11;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 10] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(sum_{k=1..n} 1/k^10).

G.f. for rationals Zeta(10, n): polylogarithm(10, x)/(1-x).

cp's OEIS Frontend

A238167 Decimal expansion of sum_(n>=1) H(n,3)/n^5 where H(n,3) = A007408(n)/A007409(n) is the n-th harmonic number of order 3.

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A244665 Decimal expansion of sum_(n>=1) (H(n,3)/n^3) where H(n,3) = A007408(n)/A007409(n) is the n-th harmonic number of order 3.

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

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A007408 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3.

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A103345 Numerator of Sum_{k=1..n} 1/k^6 = Zeta(6,n).

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A103347 Numerators of Sum_{k=1..n} 1/k^7 = Zeta(7,n).

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A103349 Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).