A007410 -id:A007410 - cp's OEIS frontend

A001008 a(n) = numerator of harmonic number H(n) = Sum_{i=1..n} 1/i.

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

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.
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) is the numerator of the polynomial A[1, n](1) where the polynomial A[genus 1, level n](m) is defined to be Sum_{d = 1..n - 1} m^(n - d)/d. (See the Mathematica procedure generating A[1, n](m) below.) - 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
a(n) coincides with A175441(n) if and only if n is not from the sequence A256102. The quotient a(n) / A175441(n) for n in A256102 is given as corresponding entry of A256103. - Wolfdieter Lang, Apr 23 2015
For a very short proof that the Harmonic series diverges, see the Goldmakher link. - N. J. A. Sloane, Nov 09 2015
All terms are odd while corresponding denominators (A002805) are all even for n > 1 (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, ... ].
Coincidences with A175441: the first 19 entries coincide because 20 is the first entry of A256102. Indeed, a(20)/A175441(20) = 55835135 / 11167027 = 5 = A256103(1). - _Wolfdieter Lang_, Apr 23 2015

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 258-261.
H. W. Gould, Combinatorial Identities, Morgantown Printing and Binding Co., 1972, # 1.45, page 6, #3.122, page 36.
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.
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.
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..2295 (first 200 terms provided by T. D. Noe)
David H. Bailey, Jonathan M. Borwein, and Roland Girgensohn, Experimental evaluation of Euler sums, Exper. Math. 3(1) (1994), 17-30; they evaluate the 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.
Hongwei Chen, Interesting Series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers, J. Int. Seq. 19 (2016), #16.1.5.
R. M. Dickau, Harmonic numbers and the book-stacking problem.
Leo Goldmakher, A short(er) proof of the divergence of the harmonic series.
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.
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv preprint arXiv:1111.3057 [math.NT], 2011.
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Hisanori Mishima, Factorizations of Wolstenholme numbers, n=1..100, n=101..200, n=201..300.
Mike Paterson et al., Maximum Overhang.
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.
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.
Roberto Tauraso, Some Congruences for Central Binomial Sums Involving Fibonacci and Lucas Numbers, JIS 19 (2016) # 16.5.4.
Eric Weisstein's World of Mathematics, Book Stacking Problem, Wolstenholme's Theorem, Harmonic Mean, Digamma Function.
Wikipedia, Harmonic number.

Cf. A002805 (denominators), A007406, A007408, A007410, A075135, A001220, A125854, A121999, A014566, A056903, A067657, A177427, A177690.
Cf. A145609-A145640. - Artur Jasinski, Oct 16 2008
Cf. A003506. - Paul Curtz, Nov 30 2013
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.
Cf. A195505.

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

import Data.Ratio ((%), numerator)
a001008 = numerator . sum . map (1 %) . enumFromTo 1
a001008_list = map numerator $ scanl1 (+) $ map (1 %) [1..]
-- Reinhard Zumkeller, Jul 03 2012

[Numerator(HarmonicNumber(n)): n in [1..30]]; // Bruno Berselli, Feb 17 2016

A001008 := proc(n)
    add(1/k,k=1..n) ;
    numer(%) ;
end proc:
seq( A001008(n),n=1..40) ; # Zerinvary Lajos, Mar 28 2007; R. J. Mathar, Dec 02 2016

Table[Numerator[HarmonicNumber[n]], {n, 30}]
(* Procedure generating A[1,n](m) (see Comments section) *) m =1; 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 *)
Numerator[Accumulate[1/Range[25]]] (* Alonso del Arte, Nov 21 2018 *)
Numerator[Table[((n - 1)/2)*HypergeometricPFQ[{1, 1, 2 - n}, {2, 3}, 1] + 1, {n, 1, 29}]] (* Artur Jasinski, Jan 08 2021 *)

A001008(n) = numerator(sum(i=1,n,1/i)) \\ Michael B. Porter, Dec 08 2009

H1008=List(1); A001008(n)={for(k=#H1008,n-1,listput(H1008,H1008[k]+1/(k+1))); numerator(H1008[n])} \\ about 100x faster for n=1..1500. - M. F. Hasler, Jul 03 2019

from sympy import Integer
[sum(1/Integer(i) for i in range(1, n + 1)).numerator() for n in range(1, 31)]  # Indranil Ghosh, Mar 23 2017

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

H(n) ~ log n + gamma + O(1/n). [See 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_{i = 1..n} Sum_{j = 1..n} 1/(i*j)). - Alexander Adamchuk, Oct 24 2004
a(n) is the numerator of Gamma/n + Psi(1 + n)/n = Gamma + Psi(n), where Psi is the digamma function. - Artur Jasinski, Nov 02 2008
H(n) = 3/2 + 2*Sum_{k = 0..n-3} binomial(k+2, 2)/((n-2-k)*(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_{k = 0..n-1} (-1)^k*binomial(n-1,k)/(k+1)^2. (Wenchang Chu) - Gary Detlefs, Apr 13 2013
H(n) = (1/2)*Sum_{k = 1..n} (-1)^(k-1)*binomial(n,k)*binomial(n+k, k)/k. (H. W. Gould) - Gary Detlefs, Apr 13 2013
E.g.f. for H(n) = a(n)/A002805(n): (gamma + log(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 is the Euler-Mascheroni constant. - Jean-François Alcover, Feb 19 2014
H(n) = Sum_{m >= 1} n/(m^2 + n*m) = gamma + digamma(1+n), numerators and denominators. (see Mathworld link on Digamma). - Richard R. Forberg, Jan 18 2015
H(n) = (1/2) Sum_{j >= 1} Sum_{k = 1..n} ((1 - 2*k + 2*n)/((-1 + k + j*n)*(k + j*n))) + log(n) + 1/(2*n). - Dimitri Papadopoulos, Jan 13 2016
H(n) = (n!)^2*Sum_{k = 1..n} 1/(k*(n-k)!*(n+k)!). - Vladimir Kruchinin, Mar 31 2016
a(n) = Stirling1(n+1, 2) / gcd(Stirling1(n+1, 2), n!) = A000254(n) / gcd(A000254(n), n!). - Max Alekseyev, Mar 01 2018
From Peter Bala, Jan 31 2019: (Start)
H(n) = 1 + (1 + 1/2)*(n-1)/(n+1) + (1/2 + 1/3)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/3 + 1/4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ... .
H(n)/n  = 1 + (1/2^2 - 1)*(n-1)/(n+1) + (1/3^2 - 1/2^2)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/4^2 - 1/3^2)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ... .
For odd n >= 3, (1/2)*H((n-1)/2) = (n-1)/(n+1) + (1/2)*(n-1)*(n-3)/((n+1)*(n+3)) + 1/3*(n-1)*(n-3)*(n-5)/((n+1)*(n+3)*(n+5)) + ... . Cf. A195505. See the Bala link in A036970. (End)
H(n) = ((n-1)/2) * hypergeom([1,1,2-n], [2,3], 1) + 1. - Artur Jasinski, Jan 08 2021
Conjecture: for nonzero m, H(n) = (1/m)*Sum_{k = 1..n} ((-1)^(k+1)/k) * binomial(m*k,k)*binomial(n+(m-1)*k,n-k). The case m = 1 is well-known; the case m = 2 is given above by Detlefs (dated Apr 13 2013). - Peter Bala, Mar 04 2022
a(n) = the (reduced) numerator 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
H(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(n,k)/k (H. W. Gould). - Gary Detlefs, May 28 2024

Edited by Max Alekseyev, Oct 21 2011

Changed title, deleting the incorrect name "Wolstenholme numbers" which conflicted with the definition of the latter in both Weisstein's World of Mathematics and in Wikipedia, as well as with OEIS A007406. - Stanislav Sykora, Mar 25 2016

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

A099828 Numerator of the generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5.

Original entry on oeis.org

1, 33, 8051, 257875, 806108207, 268736069, 4516906311683, 144545256245731, 105375212839937899, 105376229094957931, 16971048697474072945481, 16971114472329088045481, 6301272372663207205033976933

Offset: 1

Alexander Adamchuk, Oct 27 2004

From Alexander Adamchuk, Nov 07 2006: (Start)
a(n) is prime for n = {23, 25, 85, 147, 167, ...}.
There is a Wolstenholme-like theorem: p divides a(p-1) for prime p and p^2 divides a(p-1) for prime p > 7.
Also, p^3 divides a(p-1) for prime p = 5; p divides a((p-1)/2) for prime p = 37; p divides a((p-1)/3) for prime p = 37; p divides a((p-1)/4) for prime p = 37; p divides a((p-1)/5) for prime p = 11; p^2 divides a((p-1)/6) for prime p = 37; p divides a((p+1)/4) for prime p = 83; p divides a((p+1)/5) for prime p = 29; and p divides a((p+1)/6) for prime p = 11. (End)
See the Wolfdieter Lang link for information about Zeta(k, n) = H(n, k) with the rationals for k = 1..10, g.f.s, and polygamma formulas. - Wolfdieter Lang, Dec 03 2013

			H(n,5) = {1, 33/32, 8051/7776, 257875/248832, ... } = A099828/A069052.
For example, a(2) = numerator(1 + 1/2^5) = numerator(33/32) = 33 and a(3) = numerator(1 + 1/2^5 + 1/3^5) = numerator(8051/7776) = 8051. [Edited by _Petros Hadjicostas_, May 10 2020]

Alexander Adamchuk, Nov 07 2006, Table of n, a(n) for n = 1..100
Wolfdieter Lang, Rational Zeta(k,n) and more.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.

Denominators are A069052.
A099827 = H(n,5) multiplied by (n!)^5.
Cf. A001008, A007406, A007408, A007410.

Numerator[Table[Sum[1/k^5, {k, 1, n}], {n, 1, 20}]]
Numerator[Table[HarmonicNumber[n, 5], {n, 1, 20}]]
Table[Numerator[Sum[1/k^5,{k,1,n}]],{n,1,100}] (* Alexander Adamchuk, Nov 07 2006 *)

a(n) = numerator(sum(k=1, n, 1/k^5)); \\ Michel Marcus, May 10 2020

a(n) = numerator(Sum_{k=1..n} 1/k^5) = numerator(HarmonicNumber[n, 5]).

A120296 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^4.

Original entry on oeis.org

1, 15, 1231, 19615, 12280111, 4090037, 9824498837, 157151464517, 38193952437631, 7637983935923, 111835788321880643, 111830093529238643, 3194097388508809394723, 3194009594644356242723, 15970381078317764649391

Offset: 1

Alexander Adamchuk, Jul 10 2006

p divides a(p-1) for prime p > 2 - similar to Wolstenholme's theorem for A007406(n) (= numerator of Sum_{k=1..n} 1/k^2) and for A007410(n) (= numerator of Sum_{k=1..n} 1/k^4).

Lim_{n -> infinity} a(n)/A334585(n) = A267315 = (7/8)*A013662. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 15/16, 1231/1296, 19615/20736, 12280111/12960000, 4090037/4320000, 9824498837/10372320000, ... = A120296/A334585. - _Petros Hadjicostas_, May 06 2020

Cf. A007406, A007410, A013662, A119682, A267315, A334585 (denominators).

Numerator[Table[Sum[(-1)^(k+1)/k^4,{k,1,n}],{n,1,20}]]

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^4)); \\ Michel Marcus, May 07 2020

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/k^4).

Name edited by Petros Hadjicostas, May 07 2020

A007480 a(n) = denominator of sum_{k=1..n} k^(-4).

Original entry on oeis.org

1, 16, 1296, 20736, 12960000, 12960000, 31116960000, 497871360000, 40327580160000, 40327580160000, 590436101122560000, 590436101122560000, 16863445484161436160000, 16863445484161436160000, 16863445484161436160000, 269815127746582978560000

Offset: 1

N. J. A. Sloane, Mira Bernstein

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
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.

Cf. A007410.

Denominator[Accumulate[1/Range[20]^4]] (* Harvey P. Dale, Dec 25 2013 *)

a(n) = denominator of (Pi^4/90 - zeta(4, n+1)). - Arkadiusz Wesolowski, Nov 23 2012

Denominators of coefficients in expansion of PolyLog(4, x)/(1 - x). - Ilya Gutkovskiy, Apr 10 2017

A136675 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^3.

Original entry on oeis.org

1, 7, 197, 1549, 195353, 194353, 66879079, 533875007, 14436577189, 14420574181, 19209787242911, 19197460851911, 42198121495296467, 6025866788581781, 6027847576222613, 48209723660000029, 236907853607882606477

Offset: 1

Alexander Adamchuk, Jan 16 2008

a(n) is prime for n in A136683.

Lim_{n -> infinity} a(n)/A334582(n) = A197070. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 7/8, 197/216, 1549/1728, 195353/216000, 194353/216000, 66879079/74088000, 533875007/592704000, ... = a(n)/A334582(n). - _Petros Hadjicostas_, May 06 2020

Robert Israel, Table of n, a(n) for n = 1..768
Eric Weisstein's World of Mathematics, Harmonic Number.
Wikipedia, Dirichlet eta function.

Cf. A001008, A007406, A007408, A007410, A058313, A099828, A103345, A119682, A120296, A136676, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A197070, A334582 (denominators).

map(numer,ListTools:-PartialSums([seq((-1)^(k+1)/k^3, k=1..100)])); # Robert Israel, Nov 09 2023

(* Program #1 *) Table[Numerator[Sum[(-1)^(k+1)/k^3, {k,1,n}]], {n,1,50}]
(* Program #2 *) Numerator[Accumulate[Table[(-1)^(k+1) 1/k^3, {k,50}]]] (* Harvey P. Dale, Feb 12 2013 *)

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

A136677 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^6.

Original entry on oeis.org

1, 63, 45991, 2942695, 45982595359, 5109066151, 601081707598999, 38469080386820311, 252396118308232060471, 252395862211967012407, 447134922152359540530757327, 447134770212444455649757327, 2158234586764514215343657417779543, 308319185132349039219686748825649

Offset: 1

Alexander Adamchuk, Jan 16 2008

p divides a(p-1) for prime p > 2. a(n) is prime for n = {19, 47, 164, ...} = A136686.

Lim_{n -> infinity} a(n)/A334605(n) = A275703 = (31/32)*A013664. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 63/64, 45991/46656, 2942695/2985984, 45982595359/46656000000, 5109066151/5184000000, ... = a(n)/A334605(n). - _Petros Hadjicostas_, May 07 2020

Harvey P. Dale, Table of n, a(n) for n = 1..382
Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A013664, A058313, A119682, A120296, A136675, A136676.
Cf. A001008, A007406, A007408, A007410, A099828, A103345, A275703.
Cf. A136681, A136682, A136683, A136684, A136685, A136686, A334605 (denominators).

Table[ Numerator[ Sum[ (-1)^(k+1)/k^6, {k,1,n} ] ], {n,1,30} ]
Accumulate[Table[(-1)^(k+1)/k^6,{k,20}]]//Numerator (* Harvey P. Dale, Aug 21 2023 *)

A136676 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^5.

Original entry on oeis.org

1, 31, 7565, 241837, 755989457, 755889457, 12705011703799, 406547611705943, 98792790681344149, 98791774426324117, 15910615688635928566967, 15910549913780913466967, 5907492176026179821253778331

Offset: 1

Alexander Adamchuk, Jan 16 2008

a(n) is prime for n in A136685.

Lim_{n -> infinity} a(n)/A334604(n) = A267316 = (15/16)*A013663. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 31/32, 7565/7776, 241837/248832, 755989457/777600000, 755889457/777600000, ... = a(n)/A334604(n). - _Petros Hadjicostas_, May 07 2020

Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A001008, A007406, A007408, A007410, A013663, A058313, A099828, A103345, A119682, A120296, A136675, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A267316, A334604 (denominators).

Table[ Numerator[ Sum[ (-1)^(k+1)/k^5, {k,1,n} ] ], {n,1,30} ]

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^5)); \\ Michel Marcus, May 07 2020

A119722 Numerator of generalized harmonic number H(p-1,p)= Sum[ 1/k^p, {k,1,p-1}] divided by p^3 for prime p>3.

Original entry on oeis.org

2063, 2743174627, 19563315706517008974432827112201617, 2597378078067393746941400113704449589199274012223316613, 777478358612529699991463948563778410644748121498526065585976638854277886379480749840301120148933

Offset: 3

Alexander Adamchuk, Jun 13 2006

Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,p) is divisible by p^3 for prime p>3.

			Prime[3] = 5.
a(3) = numerator[ 1 + 1/2^5 + 1/3^5 + 1/4^5 ] / 5^3 = 257875/125 = 2063.
Prime[4] = 7
a(4) = numerator[ 1 + 1/2^7 + 1/3^7 + 1/4^7 + 1/5^7 + 1/6^7 ] / 7^3 = 2743174627.

Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A099828, A099827, A001008, A007406, A007408, A007410.

Numerator[Table[Sum[1/k^Prime[n],{k,1,Prime[n]-1}],{n,3,9}]]/Table[Prime[n]^3,{n,3,9}]

a(n) = numerator[ Sum[ 1/k^Prime[n], {k,1,Prime[n]-1} ]] / Prime[n]^3 for n>2.

A136681 Numbers k such that A058313(k) is prime.

Original entry on oeis.org

3, 4, 5, 6, 9, 10, 13, 16, 17, 18, 37, 43, 58, 121, 124, 126, 137, 203, 247, 283, 285, 286, 289, 317, 424, 508, 751, 790, 937, 958, 1066, 1097, 1151, 1166, 1194, 1199, 1235, 1414, 1418, 1460, 1498, 1573, 2090, 2122, 2691, 2718, 3030, 3426, 3600, 3653, 3737

Offset: 1

Alexander Adamchuk, Jan 16 2008

nonn

A058313(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A058313, A119682, A120296, A136675, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136682, A136683, A136684, A136685, A136686.

Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k, {k,1,n} ] ]; If[ PrimeQ[f], Print[ {n,f} ] ], {n,1,317} ]

isok(n) = isprime(numerator(sum(k=1, n, (-1)^(k+1)/k))); \\ Michel Marcus, Mar 14 2019

a(25)-a(30) from James R. Buddenhagen, Sep 22 2015

a(31)-a(51) from Amiram Eldar, Mar 14 2019

cp's OEIS Frontend

A001008 a(n) = numerator of harmonic number H(n) = Sum_{i=1..n} 1/i.

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

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A099828 Numerator of the generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5.

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A120296 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^4.

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A007480 a(n) = denominator of sum_{k=1..n} k^(-4).

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A136675 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^3.

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A136677 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^6.