A058312 -id:A058312 - cp's OEIS frontend

A119248 a(n) is the difference between denominator and numerator of the n-th alternating harmonic number Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n).

0, 1, 1, 5, 13, 23, 101, 307, 641, 893, 7303, 9613, 97249, 122989, 19793, 48595, 681971, 818107, 13093585, 77107553, 66022193, 76603673, 1529091919, 1752184789, 7690078169, 8719737569, 23184641107, 3721854001, 96460418429

Offset: 1

Alexander Adamchuk, Jul 22 2006

a(n)/A058312(n) = 1 - A058313(n)/A058312(n) appears in the locker puzzle (see the links in A364317) for the probability of success with the strategy used there for n lockers and allowed openings of up to floor(n/2) lockers. Note that gcd(a(n), A058312(n)) = 1. - Wolfdieter Lang, Aug 12 2023

Cf. A058312, A058313, A075829, A364317.

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

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

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/k) - numerator(Sum_{k=1..n} (-1)^(k+1)/k).
a(n) = A058312(n) - A058313(n).
a(n) = A075829(n+1).
a(n) = numerator(Sum_{k=2..n} (-1)^k/k). (Cf. A024168.) - Petros Hadjicostas, May 17 2020

A269626 a(n) = A003418(n) / A058312(n) with a(0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 55, 55, 55, 55, 55, 55, 55

Offset: 0

Jeppe Stig Nielsen, Mar 01 2016

nonn

Cf. A003418, A058312, A110566.

a(n)=lcm([1..n])/denominator(sum(k=1,n,(-1)^(k+1)/k))

A337920 Numbers k such that d(k) = d(k+1), where d(k) = A058312(k) is the denominator of the k-th alternating harmonic number.

Original entry on oeis.org

5, 9, 11, 13, 17, 20, 21, 23, 25, 29, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 75, 77, 79, 81, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 97, 98, 101, 104, 105, 107, 109, 110, 111, 113

Offset: 1

Amiram Eldar, Jan 29 2021

nonn

The asymptotic density of this sequence is 1 (Wu and Chen, 2019).

			5 is a term since A058312(5) = A058312(6) = 60.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bing-Ling Wu and Yong-Gao Chen, On the denominators of harmonic numbers, II, Journal of Number Theory, Vol. 200 (2019), pp. 397-406.

Cf. A058312, A065454 (analogous with denominators of harmonic numbers).

d[n_] := Denominator @ Sum[(-1)^(k + 1)/k, {k, 1, n}]; Position[Partition[d[Range[120]], 2, 1], {x_, x_}] // Flatten

A351789 Decimal expansion of Sum_{k>=1} AH(k)*F(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 19 2022

			1.51437037420622187243459478916165077964831313316887...

Seán M. Stewart, Problem H-893, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 60, No. 1 (2022), p. 91.

Cf. A000045, A001622, A002390, A058312, A058313, A349850, A351794.

RealDigits[Log[5/4] + 6*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]

Equals log(5/4) + 6*log(phi)/sqrt(5), where phi is the golden ratio (A001622) (Stewart, 2022).

A351794 Decimal expansion of Sum_{k>=1} AH(k)*L(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and L(k) = A000032(k) is the k-th Lucas number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 19 2022

			2.82147535876264946174605143568253063724665666934546...

Cf. A000032, A001622, A002390, A058312, A058313, A349850, A349851, A351789.

RealDigits[3*Log[5/4] + 10*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]

Equals 3*log(5/4) + 10*log(phi)/sqrt(5), where phi is the golden ratio (A001622).

Equals (4/3)*log(5/4) + (5/3)*A351789.

A058313 Numerator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.

Original entry on oeis.org

1, 1, 5, 7, 47, 37, 319, 533, 1879, 1627, 20417, 18107, 263111, 237371, 52279, 95549, 1768477, 1632341, 33464927, 155685007, 166770367, 156188887, 3825136961, 3602044091, 19081066231, 18051406831, 57128792093, 7751493599, 236266661971

Offset: 1

N. J. A. Sloane, Dec 09 2000

A Wolstenholme-like theorem: for prime p > 3, if p = 6k-1, then p divides a(4k-1), otherwise if p = 6k+1, then p divides a(4k). - T. D. Noe, Apr 01 2004
For the limit n -> infinity of the partial sums of the alternating harmonic series see A002162. - Wolfdieter Lang, Sep 08 2015
a(n)/A058312(n) appears in the locker puzzle (see the links in A364317) as the probability of failures with the strategy used for n lockers and opening of up to floor(n/2) lockers. Note the alternative formula given below for a(n)/A058312(n) using only positive fractions. - Wolfdieter Lang, Aug 12 2023

			1, 1/2, 5/6, 7/12, 47/60, 37/60, 319/420, 533/840, 1879/2520, ...
For n=4: a(n)/A058312(n) = 7/12 because 1/1 - 1/2 + 1/3 - 1/4 = 7/12 = 1/4 + 1/3. - _Wolfdieter Lang_, Aug 12 2023

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, page 14, #71.

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..2000 (terms up to a(200) from T. D. Noe)
Khristo N. Boyadzhiev, Power series with skew-harmonic numbers, dilogarithms, and double integrals, Tatra Mt. Math. Publ., Vol. 56 (2013), pp. 93-108.
Hisanori Mishima, Factorizations of many number sequences.
Hisanori Mishima, Factorizations of many number sequences.
Michael Penn, A number theory problem from an unlikely source, YouTube video, 2025.
Eric Weisstein's World of Mathematics, Harmonic Number.

Denominators are A058312. Cf. A025530.
Apart from leading term, same as A075830.
Cf. A001008 (numerator of n-th harmonic number).
Bisections are A049281 and A082687.
Cf. A181983.

import Data.Ratio((%), numerator)
a058313 n = a058313_list !! (n-1)
a058313_list = map numerator $ scanl1 (+) $ map (1 %) $ tail a181983_list
-- Reinhard Zumkeller, Mar 20 2013

A058313 := n->numer(add((-1)^(k+1)/k,k=1..n));
# Alternatively:
a := n -> numer(harmonic(n) - harmonic((n-modp(n,2))/2)):
seq(a(n), n=1..29); # Peter Luschny, May 03 2016

Numerator[Table[Sum[(-1)^(k + 1)/k, {k, n}], {n, 30}]] (* Harvey P. Dale, Jul 18 2012 *)
a[n_]:= (-1)^n (HarmonicNumber[n/2 - 1/2] - HarmonicNumber[n/2] + (-1)^n Log[4])/2; Table[a[n] // FullSimplify, {n, 29}] // Numerator (* Gerry Martens, Jul 05 2015 *)
Rest[Numerator[CoefficientList[Series[Log[1 + x]/(1 - x), {x, 0, 33}], x]]] (* Vincenzo Librandi, Jul 06 2015 *)
Table[Log[2] - (-1)^n LerchPhi[-1, 1, n + 1], {n, 20}] // Numerator (* Eric W. Weisstein, Aug 25 2023 *)

a(n)=(-1)^n*numerator(polcoeff(log(1-x)/(x+1)+O(x^(n+1)), n))

G.f. for a(n)/A058312(n): log(1+x)/(1-x). - Benoit Cloitre, Jun 15 2003
a(n) = (n*a(n-1) + (-1)^(n+1)*A058312(n-1))/gcd(n*a(n-1) + (-1)^(n+1)*A058312(n-1), n*A058312(n-1)). - Robert Israel, Jul 06 2015
From Peter Luschny, May 03 2016: (Start)
Let H(n) denote the harmonic numbers, AH(n) denote the alternating harmonic numbers, Psi the polygamma function and euler(n,x) the Euler polynomials. Then:
AH(n) = H(n) - H((n - n mod 2)/2).
AH(z) = log(2)+(1/2)*cos(Pi*z)*(Psi(z/2+1/2)-Psi(z/2+1)).
AH(z) ~ log(2)+(1/2)*cos(Pi*z)*(-1/z+1/(2*z^2)-1/(4*z^4)+1/(2*z^6)-...).
AH(z) ~ log(2)-(1/2)*cos(Pi*z)*Sum_{n>=0} Euler(n,0)/z^(n+1). (End)
Sum_{k>=1} (-1)^(k+1)*AH(k)/k = Pi^2/12 + log(2)^2/2 (Boyadzhiev, 2013). - Amiram Eldar, Oct 04 2021
a(n)/A058312(n) = Sum_{j=0..ceiling(n/2) - 1} 1/(n-j), for n >= 1. (Proof by comparing the recurrences for even and odd n.) - Wolfdieter Lang, Aug 12 2023
For n >= 1, log(2) = a(n)/A058312(n) + (-1)^n*n!*Sum_{k >= 1} 1/(k*(k + 1)* ...*(k + n)*2^k). - Peter Bala, Dec 07 2023
a(n) = the (reduced) numerator of the continued fraction 1/(1 + 1^2/(1 + 2^2/(1 + 3^2/(1 + ... + (n-1)^2/(1))))). - Peter Bala, Feb 18 2024

A197070 Decimal expansion of the Dirichlet eta-function at 3.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 09 2011

This constant is irrational by Apéry's theorem. - Charles R Greathouse IV, Feb 11 2024

			0.9015426773696957140498036211335874930737...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
R. Barbieri, J. A. Mignaco and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864 Table II (4)
Su Hu, Min-soo Kim, Euler's integral, multiple cosine function and zeta values, arXiv:2201.011247 (2023), series last equation.
Seán Stewart, Problem 12206, The American Mathematical Monthly, Vol. 127, No. 8 (2020), p. 752.
Wikipedia, Dirichlet eta function.

Cf. A002117 (zeta(3)), A058312, A058313, A072691, A136675, A233090 (5*zeta(3)/8), A233091 (7*zeta(3)/8), A334582.

Maple
```
3*Zeta(3)/4 ; evalf(%) ;
```

RealDigits[3(Zeta[3])/4, 10, 75][[1]] (* Bruno Berselli, Dec 20 2011 *)

-polylog(3,-1) \\ Charles R Greathouse IV, Mar 28 2012

3/4*zeta(3) \\ Charles R Greathouse IV, Mar 28 2012

Equals 3*zeta(3)/4 = 3*A002117/4.
Also equals the integral over the unit cube [0,1]x[0,1]x[0,1] of 1/(1+x*y*z) dx dy dz. - Jean-François Alcover, Nov 24 2014
Equals Sum_{n>=1} (-1)^(n+1)/n^3. - Terry D. Grant, Aug 03 2016
Equals Lim_{n -> infinity} A136675(n)/A334582(n). - Petros Hadjicostas, May 07 2020
Equals Sum_{n>=1} AH(2*n)/n^2, where AH(n) = Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n) is the n-th alternating harmonic number (Stewart, 2020). - Amiram Eldar, Oct 04 2021
Equals -int_0^1 log(x)log(1+x)/x dx [Barbieri] - R. J. Mathar, Jun 07 2024

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

Original entry on oeis.org

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)).

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

cp's OEIS Frontend

A119248 a(n) is the difference between denominator and numerator of the n-th alternating harmonic number Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n).

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A269626 a(n) = A003418(n) / A058312(n) with a(0) = 1.

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A337920 Numbers k such that d(k) = d(k+1), where d(k) = A058312(k) is the denominator of the k-th alternating harmonic number.

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A351789 Decimal expansion of Sum_{k>=1} AH(k)*F(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

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A351794 Decimal expansion of Sum_{k>=1} AH(k)*L(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and L(k) = A000032(k) is the k-th Lucas number.

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A058313 Numerator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.

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A197070 Decimal expansion of the Dirichlet eta-function at 3.

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

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