A058313 -id:A058313 - cp's OEIS frontend

A288965 Number of key comparisons to sort all n! permutations of n elements by the optimal dual-pivot quicksort.

0, 0, 2, 16, 114, 866, 7188, 65580, 655872, 7157376, 84775680, 1084343040, 14906039040, 219267751680, 3437854963200, 57247256424960, 1009189972869120, 18779054120386560, 367876307230064640, 7568437652936294400, 163164173556503347200, 3678547646214424166400

Offset: 0

Daniel Krenn, Jun 20 2017

nonn

Daniel Krenn, Table of n, a(n) for n = 0..100
M. Aumüller, M. Dietzfelbinger, C. Heuberger, D. Krenn, and H. Prodinger, Dual-Pivot Quicksort: Optimality, Analysis and Zeros of Associated Lattice Paths, arXiv:1611.00258 [math.CO], 2016.
M. Aumüller, M. Dietzfelbinger, C. Heuberger, D. Krenn, and H. Prodinger, Dual-Pivot Quicksort: Optimality, Analysis and Zeros of Associated Lattice Paths, Combin. Probab. Comput. 28(4) (2019), 485-518.
Daniel Krenn, Quickstar, Program in SageMath, on GitHub.
R. Neininger and J. Straub, Probabilistic analysis of the dual-pivot quicksort "count", arXiv:1710.07505 [cs.DS], 2017.
R. Neininger and J. Straub, Probabilistic analysis of the dual-pivot quicksort "count", 2018 Proceedings of the Meeting of Analytic Algorithmics and Combinatorics, New Orleans, Louisiana, USA, 7pp.
Index entries for sequences related to sorting

Cf. A058312, A058313, A288964, A288970, A288971.

haralt := proc(n) local k: add((-1)^k/k, k = 1 .. n): end proc:
a := proc(n) local v, v1, v2: if n = 0 or n = 1 then v := 0: end if; if n = 2 then v := 2: end if: if n = 3 then v := 16: end if:
if 4 <= n then v1 := 9/5*n*harmonic(n) - 1/5*n*haralt(n) - 89/25*n + 67/40*harmonic(n) - 3/40*haralt(n) - 83/800 + 1/10*(-1)^n:
if 0 = n mod 2 then v2 := -1/320*1/(n - 3) - 3/320*1/(n - 1): end if:
if 1 = n mod 2 then v2 := 3/320*1/(n - 2) + 1/320*1/n: end if:
v := n!*(v1 + v2): end if: v: end proc:
seq(a(n1), n1 = 0 .. 30); # Petros Hadjicostas, May 03 2020

haralt[n_] := Sum[(-1)^k/k, {k, 1, n}];
a[n_] := Switch[n, 0|1, 0, 2, 2, 3, 16, _, n! ((9/5) n HarmonicNumber[n] - (1/5) n haralt[n] - (89/25) n + (67/40) HarmonicNumber[n] - (3/40)* haralt[n] - 83/800 + (-1)^n/10 - (Boole[EvenQ[n]]/320)(1/(n-3) + 3/(n-1)) + (Boole[OddQ[n]]/320)(3/(n-2) + 1/n))];
a /@ Range[0, 21] (* Jean-François Alcover, Jun 05 2020 *)

lista(nn) = {my(x='x + O('x^nn)); concat([0,0],Vec(serlaplace(-8*log(1-x)/(5*(1-x)^2) + 2*atanh(x)/(5*(1-x)^2) - 44/(25*(1-x)^2) - atanh(x)/(4*(1-x)) + 281/(160*(1-x)) + ((1-x)^3/320)*atanh(x) + x^3/150 - 27*x^2/1600 + 17*x/1600 + 3/800)))}; \\ Petros Hadjicostas and Michel Marcus, May 04 2020, e.g.f. from p. 29 in Aumüller et al. (2016)

a(n) = n!*((9/5)*n*Harmonic(n) - (1/5)*n*Harmonic_alt(n) - (89/25)*n + (67/40)*Harmonic(n) - (3/40)*Harmonic_alt(n) - 83/800 + (-1)^n/10 - ([n even]/320)*(1/(n - 3) + 3/(n - 1)) + ([n odd]/320)*(3/(n - 2) + 1/n)) for n >= 4, where [condition] = 1 if the condition holds, and 0 otherwise, and Harmonic_alt(n) = Sum_{k=1..n} (-1)^n/n = -A058313(n)/A058312(n). [This follows from Theorem 12.1 in Aumüller et al. (2016) or Theorem 5.7 in Aumüller et al. (2019).] - Petros Hadjicostas, May 03 2020

A082687 Numerator of Sum_{k=1..n} 1/(n+k).

Original entry on oeis.org

1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 7751493599, 225175759291, 13981692518567, 14000078506967, 98115155543129, 3634060848592973, 3637485804655193

Offset: 1

Benoit Cloitre, Apr 12 2003

Numerator of Sum_{k=0..n-1} 1/((k+1)(2k+1)) (denominator is A111876). - Paul Barry, Aug 19 2005
Numerator of the sum of all matrix elements of n X n Hilbert matrix M(i,j) = 1/(i+j-1) (i,j = 1..n). - Alexander Adamchuk, Apr 11 2006
Numerator of the 2n-th alternating harmonic number H'(2n) = Sum ((-1)^(k+1)/k, k=1..2n). H'(2n) = H(2n) - H(n), where H(n) = Sum_{k=1..n} 1/k is the n-th Harmonic Number. - Alexander Adamchuk, Apr 11 2006
a(n) almost always equals A117731(n) = numerator(n*Sum_{k=1..n} 1/(n+k)) = numerator(Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1)) but differs for n = 14, 53, 98, 105, 111, 114, 119, 164. - Alexander Adamchuk, Jul 16 2006
Sum_{k=1..n} 1/(n+k) = n!^2 *Sum_{j=1..n} (-1)^(j+1) /((n+j)!(n-j)!j). - Leroy Quet, May 20 2007
Seems to be the denominator of the harmonic mean of the first n hexagonal numbers. - Colin Barker, Nov 19 2014
Numerator of 2*n*binomial(2*n,n)*Sum_{k = 0..n-1} (-1)^k* binomial(n-1,k)/(n+k+1)^2. Cf. A049281. - Peter Bala, Feb 21 2017
From Peter Bala, Feb 16 2022: (Start)
2*Sum_{k = 1..n} 1/(n+k) = 1 + 1/(1*2)*(n-1)/(n+1) - 1/(2*3)*(n-1)*(n-2)/((n+1)*(n+2)) + 1/(3*4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + - .... Cf. A101028.
2*Sum_{k = 1..n} 1/(n+k) = n - (1 + 1/2^2)*n*(n-1)/(n+1) + (1/2^2 + 1/3^2)*n*(n-1)*(n-2)/((n+1)*(n+2)) - (1/3^2 + 1/4^2)*n*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + (1/4^2 + 1/5^2)*n*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) - + .... Cf. A007406 and A120778.
These identities allow us to extend the definition of Sum_{k = 1..n} 1/(n+k) to non-integral values of n. (End)

			H'(2n) = H(2n) - H(n) = {1/2, 7/12, 37/60, 533/840, 1627/2520, 18107/27720, 237371/360360, 95549/144144, 1632341/2450448, 155685007/232792560, ...}, where H(n) = A001008/A002805.
n=2: HilbertMatrix(n,n)
   1  1/2
  1/2 1/3
so a(2) = Numerator(1 + 1/2 + 1/2 + 1/3) = Numerator(7/3) = 7.
The n X n Hilbert matrix begins:
   1   1/2  1/3  1/4  1/5  1/6  1/7  1/8  ...
  1/2  1/3  1/4  1/5  1/6  1/7  1/8  1/9  ...
  1/3  1/4  1/5  1/6  1/7  1/8  1/9  1/10 ...
  1/4  1/5  1/6  1/7  1/8  1/9  1/10 1/11 ...
  1/5  1/6  1/7  1/8  1/9  1/10 1/11 1/12 ...
  1/6  1/7  1/8  1/9  1/10 1/11 1/12 1/13 ...

T. D. Noe, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Bisection of A058313, A082688 (denominators).
Cf. A001008, A002805, A005249, A007406, A058312.
Cf. A086881, A098118, A101028, A117731, A120778.

[Numerator((HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // G. C. Greubel, Jul 24 2023

a := n -> numer(harmonic(2*n) - harmonic(n)):
seq(a(n), n=1..20); # Peter Luschny, Nov 02 2017

Numerator[Sum[1/k,{k,1,2*n}] - Sum[1/k,{k,1,n}]] (* Alexander Adamchuk, Apr 11 2006 *)
Table[Numerator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 20}] (* Alexander Adamchuk, Apr 11 2006 *)
Table[HarmonicNumber[2 n] - HarmonicNumber[n], {n, 20}] // Numerator (* Eric W. Weisstein, Dec 14 2017 *)

a(n) = numerator(sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017

[numerator(harmonic_number(2*n,1) - harmonic_number(n,1)) for n in range(1,41)] # G. C. Greubel, Jul 24 2023

Limit_{n -> oo} Sum_{k=1..n} 1/(n+k) = log(2).
Numerator of Psi(2*n+1) - Psi(n+1). - Vladeta Jovovic, Aug 24 2003
a(n) = numerator((Sum_{k=1..2*n} 1/k) - Sum_{k=1..n} 1/k). - Alexander Adamchuk, Apr 11 2006
a(n) = numerator(Sum_{j=1..n} (Sum_{i=1..n} 1/(i+j-1))). - Alexander Adamchuk, Apr 11 2006
The o.g.f for Sum_{k=1..n} 1/(n+k) is f(x) = (sqrt(x)*log((1+sqrt(x))/(1-sqrt(x))) + log(1-x))/(2*x*(1-x)).

A126197 GCDs arising in A126196.

Original entry on oeis.org

11, 1093, 1093, 3511, 3511, 5557, 104891, 1006003

Offset: 1

Max Alekseyev and Tanya Khovanova, Mar 07 2007

All terms are primes. Note a connection to the Wieferich primes A001220: a(2) = a(3) = A001220(1), a(3) = a(4) = A001220(2).
From John Blythe Dobson, Jan 14 2017: (Start)
All Wieferich primes p will belong to this sequence twice, because if H([p/k]) denotes the harmonic number with index floor(p/k), then p divides all of H([p/4]), H([p/2]), and H(p-1). The first two of these elements gives one solution, and the second and third another. This property of the Wieferich primes predates their name, and was apparently first proved by Glaisher in "On the residues of r^(p-1) to modulus p^2, p^3, etc.," pp. 21-22, 23 (see References).
Note also a connection to the Mirimanoff primes A014127: a(1) = A014127(1), a(8) = A014127(2). All Mirimanoff primes p will belong to this sequence, because p divides both H([p/3]) and H([2p/3]). This property of the Mirimanoff primes likewise predates their name, and was apparently first proved by Glaisher in "A general congruence theorem relating to the Bernoullian function," p. 50 (see Links).
The Wieferich primes and Mirimanoff primes would seem to be the only cases for which the value of n in A126196(n) is predictable from knowledge of p. It is not obvious that all members of the present sequence are prime; however, by definition all their divisors must be non-harmonic primes A092102. Furthermore, it is clear from the cited literature under that entry that H([n/2]) == H(n) == 0 (mod p) is only possible when n < p. Thus, all divisors of the present sequence must belong to the harmonic irregular primes A092194.
One possible reason for interest in this sequence is a 1995 result of Dilcher and Skula (see Links) which among other things shows that if a prime p were an exception to the first case of Fermat's Last Theorem, then p would divide both H([p/k]) and H([2p/k]) for every value of k from 2 to 46. To date, the only values for which such coincidences have been found have k = 2, 3, or 4. For k = 6 to hold, p would have to be simultaneously a Wieferich prime and a Mirimanoff prime, while for k = 5 to hold, p would have to be simultaneously a Wall-Sun-Sun prime and a member of A123692. The sparse numerical results for the present sequence suggest that even the more relaxed condition H([n/2]) == H(n) == 0 (mod p) is rarely satisfied. (End)

J. W. L. Glaisher, On the residues of r^(p-1) to modulus p^2, p^3, etc., Quarterly Journal of Pure and Applied Mathematics 32 (1900-1901), 1-27.

Karl Dilcher and Ladislav Skula, A new criterion for the first case of Fermat's Last Theorem, Mathematics of Computation 64 (1995), 363-392.
J. W. L. Glaisher, A general congruence theorem relating to the Bernoullian function, Proceedings of the London Mathematical Society 33 (1900-1901), 27-56.

Cf. A001008, A001220, A002805, A058313, A074791, A121594, A125581, A126196.

f[n_] := GCD @@ Numerator@ HarmonicNumber@ {n, Floor[n/2]}; f@ Select[ Range[5000], f[#] > 1 &] (* Giovanni Resta, May 13 2016 *)

a(8) from Giovanni Resta, May 13 2016

A125581 Numbers n such that n does not divide the denominator of the n-th harmonic number nor the denominator of the n-th alternating harmonic number.

Original entry on oeis.org

77, 847, 9317, 102487, 596778, 1127357, 1193556, 6161805, 12323610, 12400927

Offset: 1

Alexander Adamchuk, Jan 03 2007

Note that a(1) = 7*11, a(2) = 7*11^2, and a(3) = 7*11^3.
Harmonic numbers are defined as H(n) = Sum_{k=1..n} 1/k = A001008(n)/A002805(n).
Alternating harmonic numbers are defined as H'(n) = Sum_{k=1..n} (-1)^(k+1)*1/k = A058313(n)/A058312(n).
Numbers n such that n does not divide the denominator of the n-th harmonic number are listed in A074791. Numbers n such that n does not divide the denominator of the n-th alternating harmonic number are listed in A121594.
This sequence is the intersection of A074791 and A121594.
Comments from Max Alekseyev, Mar 07 2007: (Start)
While A125581 indeed contains the geometric progression 7*11^n as a subsequence, it also contains other geometric progressions such as: 546*1093^n, 1092*1093^n, 1755*3511^n, 3510*3511^n and 4896*5557^n (see A126196 and A126197). It may also contain some "isolated" terms (i.e. not participating in the geometric progressions) but such terms are harder to find and at the moment I have no proof that they exist.
This is a sketch of my proof that geometric progression 7*11^n and the others mentioned above belong to A125581.
Lemma 1. H'(n) = H(n) - H([n/2]).
Lemma 2. For prime p and integer n >= p, valuation(H(n),p) >= valuation(H([n/p]),p) - 1.
Theorem. For an integer b > 1 and a prime number p such that p divides the numerators of both H(b) and H([b/2]), the geometric progression b*p^n belongs to A125581.
Proof. It is enough to show that valuation(H(b*p^n),p) > -n and valuation(H'(b*p^n), p) > -n. By Lemma 2 we have valuation(H(b*p^n), p) >= valuation(H(b), p) - n >= 1 - n > -n.
From this inequality and Lemma 1, we have valuation(H'(b*p^n), p) >= min{ valuation(H(b*p^n), p), valuation(H([b*p^n/2]), p) } >= min{ 1 - n, valuation(H([b*p^n/2]), p) }. It remains to show that valuation(H([b*p^n/2]), p) >= 1 - n.
Again by Lemma 2, we have valuation(H([b*p^n/2]), p) >= valuation(H([b/2]), p) - n >= 1 - n, which completes the proof.
It is easy to check that this Theorem holds for the aforementioned geometric progressions. (End)

Tanya Khovanova, Non Recursions.
Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A001008, A002805, A003599, A058312, A058313, A074791, A119955, A121594.

f=0; g=0; Do[g=g+1/n; f=f+(-1)^(n+1)/n; If[ !IntegerQ[Denominator[g]/n]&&!IntegerQ[Denominator[f]/n], Print[n]], {n, 1, 10000}]

More terms from Max Alekseyev, Mar 11 2007

a(8)-a(10) from Max Alekseyev, Mar 19 2007

A126196 Numbers k such that gcd(A001008(k), A001008(floor(k/2))) > 1.

Original entry on oeis.org

7, 546, 1092, 1755, 3510, 4896, 52447, 670668

Offset: 1

Max Alekseyev and Tanya Khovanova, Mar 07 2007, corrected Mar 10 2007

Note a connection to the Wieferich primes A001220: a(2) = (A001220(1) - 1)/2, a(3) = A001220(1) - 1, a(4) = (A001220(2) - 1)/2, a(5) = A001220(2) - 1. [Comment regarding a(2) added by Kevin J. Gomez, Jul 11 2017]

a(9) > 840000. - Giovanni Resta, May 13 2016

The corresponding GCDs are given by A126197.

Cf. A001008, A001220, A002805, A058312, A058313, A074791, A121594, A125581.

Select[Range[5000], GCD @@ Numerator@ HarmonicNumber@{#, Floor[#/2]} > 1 &] (* Giovanni Resta, May 13 2016 *)

a001008(n)=numerator(sum(i=1, n, 1/i))
for(n=1, 1e6, if(gcd(a001008(n), a001008(n/2)) > 1, print1(n, ", "))) \\ Felix Fröhlich, Aug 08 2014

a(8) from Giovanni Resta, May 13 2016

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

A117731 Numerator of the fraction n*Sum_{k=1..n} 1/(n+k).

Original entry on oeis.org

1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 54260455193, 225175759291, 13981692518567, 14000078506967, 98115155543129, 3634060848592973, 3637485804655193

Offset: 1

Alexander Adamchuk, Apr 14 2006

a(n) almost always equals A082687(n), but differs for n in A125740.

p divides a((p-1)/3) for primes p in A002476, that is, primes of form 6*n + 1. - Alexander Adamchuk, Jul 16 2006

			The first few fractions are 1/2, 7/6, 37/20, 533/210, 1627/504, 18107/4620, 237371/51480, ... = A117731/A296519.
For n=2, the n X n Hilbert matrix is
  1 1/2
  1/2 1/3
Thus, a(2) = numerator(1 + 1/2 + 1/2 + 1/3) = numerator(7/3) = 7.
The n X n Hilbert matrix begins as follows:
    1 1/2 1/3 1/4  1/5  1/6  1/7  1/8 ...
  1/2 1/3 1/4 1/5  1/6  1/7  1/8  1/9 ...
  1/3 1/4 1/5 1/6  1/7  1/8  1/9 1/10 ...
  1/4 1/5 1/6 1/7  1/8  1/9 1/10 1/11 ...
  1/5 1/6 1/7 1/8  1/9 1/10 1/11 1/12 ...
  1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
  ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Cf. A296519 (denominators).

Cf. A001008, A002476, A005249, A058313, A058312, A082687, A086881, A098118, A117664, A125740.

[Numerator(n*(HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // G. C. Greubel, Jul 24 2023

Numerator[Table[n Sum[1/(n + k), {k, n}], {n, 1, 100}]]
Numerator[Table[Sum[Sum[1/(i + j - 1), {i, n}], {j, n}], {n, 30}]] (* Alexander Adamchuk, Apr 23 2006 *)
Table[n (HarmonicNumber[2 n] - HarmonicNumber[n]), {n, 20}] // Numerator (* Eric W. Weisstein, Dec 14 2017 *)

a(n) = numerator(n*sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017

[numerator(n*(harmonic_number(2*n,1) - harmonic_number(n,1))) for n in range(1,41)] # G. C. Greubel, Jul 24 2023

a(n) = numerator(n*Sum_{k=1..n} 1/(n+k)).
a(n) = numerator(n*(Psi(2*n+1) - Psi(n+1))).
a(n) = numerator(n*Sum_{k=1..2*n} (-1)^(k+1)/k).
a(n) = numerator(n*A058313(2*n)/A058312(2*n)).
a(n) = numerator(Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1)), which is the numerator of the sum of all matrix elements of n X n Hilbert Matrix M(i,j) = 1/(i+j-1), (i,j = 1..n). The denominator is A117664(n). - Alexander Adamchuk, Apr 23 2006

Various sections edited by Petros Hadjicostas and Michel Marcus, May 07 2020

A136682 Numbers k such that A119682(k) is prime.

Original entry on oeis.org

2, 3, 5, 8, 23, 41, 47, 48, 49, 95, 125, 203, 209, 284, 323, 395, 504, 553, 655, 781, 954, 1022, 1474, 1797, 1869, 1923, 1934, 1968, 2043, 2678, 3413, 3439, 4032, 4142, 4540, 4895, 5018, 5110, 5194, 5357, 6591, 11504, 11949, 14084, 20365

Offset: 1

Alexander Adamchuk, Jan 16 2008

nonn

A119682(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^2.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A058313, A119682, A120296, A136675, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136681, A136683, A136684, A136685, A136686.

Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^2, {k,1,n} ] ]; If[ PrimeQ[f], Print[ {n,f} ] ], {n,1,125} ]

a(12)-a(17) from Alexander Adamchuk, Apr 28 2008
a(18)-a(31) from Amiram Eldar, Mar 14 2019
a(32)-a(45) from Robert Price, Apr 14 2019

cp's OEIS Frontend

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A082687 Numerator of Sum_{k=1..n} 1/(n+k).

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A125581 Numbers n such that n does not divide the denominator of the n-th harmonic number nor the denominator of the n-th alternating harmonic number.

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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.

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