A082687 -id:A082687 - cp's OEIS frontend

A125740 Numbers n such that A117731(n) differs from A082687(n).

14, 52, 98, 105, 111, 114, 119, 164, 310, 444, 518, 602, 676, 686, 715, 735, 749, 833, 1220, 1278, 1339, 1474, 1752, 1946, 2023, 2054, 2166, 3016, 3104, 3502, 3568, 3924, 4107, 4308, 4802, 5145, 5243, 5334, 5718, 5831, 6394, 6724, 7550, 8135, 8164, 8767

Offset: 1

Alexander Adamchuk, Dec 04 2006, Mar 12 2007

nonn

All listed terms are composite.
The ratio of A117731(n) and A082687(n) when they are different is listed in A125741(n) = A117731[ a(n) ] / A082687[ a(n) ] = {7, 13, 7, 7, 37, 19, 119, 41, 31, 37, 37, 43, 13, 7, 13, 49, 7, 7, 61, 71, 103, 67, 73, 139, ...}.
It appears that all (or almost all) members of geometric progressions 2*7^k, 4*13^k, 15*7^k, 3^37^k, 6*19^k, 17*7^k, 4*41^k, 10*31^k, 12*37^k, 55*13^k, 107*7^k, etc. for k>0 are in the sequence.

			A117731(n) begins {1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 54260455193, ...}.
A082687(n) begins {1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 7751493599, ...}.
Thus a(1) = 14 because for n<14 A117731(n) = A082687(n) but A117731(14) = 54260455193 differs from A082687(14) = 7751493599.

Eric Weisstein, The World of Mathematics: Hilbert Matrix.
Eric Weisstein, The World of Mathematics: Harmonic Number

Cf. A117731 = Numerator of n*Sum[ 1/(n+k), {k, 1, n} ]. Cf. A082687 = Numerator of Sum[ 1/(n+k), {k, 1, n} ]. Cf. A125741 = The ratio of A117731(n) and A082687(n) when they are different.
Cf. A082687(n) = 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 (1/k, k=1..n) is the n-th harmonic number. A117731(n) = 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).
Cf. A126196, A126197, 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.

h=0; Do[ h=h+1/(n+1)/(2n+1); f=Numerator[n*h]; g=Numerator[h]; If[ !Equal[f,g], Print[n] ], {n,1,17381} ]

A125741 The ratio of A117731(n) and A082687(n) when they are different.

Original entry on oeis.org

7, 13, 7, 7, 37, 19, 119, 41, 31, 37, 37, 43, 13, 7, 13, 49, 7, 7, 61, 71, 103, 67, 73, 139, 17, 79, 19, 29, 97, 103, 223, 109, 37, 359, 7, 49, 7, 127, 953, 7, 139, 41, 151, 1627, 157, 797, 179, 13, 163, 13, 13, 13, 13, 13, 31, 31, 181, 193, 199, 919, 193, 211, 757, 37

Offset: 1

Alexander Adamchuk, Dec 04 2006

Corresponding numbers n such that A117731(n) differs from A082687(n) are listed in A125740(n) = {14, 52, 98, 105, 111, 114, 119, 164, 310, 444, 518, 602, 676, 686, 715, 735, 749, 833, ...}. a(n) divides A125740(n). Most a(n) are primes.

The first composite term in a(n) is a(7) = 119 = 7*17. a(n) is composite for n = {7, 16, 36}. a(16) = a(36) = 49 = 7^2.

			A082687(n) begins {1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 7751493599, ...}.
Thus a(1) = 7 because A117731(n)/A082687(n) = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1,...}.

Cf. A125740 = numbers n such that A117731(n) differs from A082687(n). Cf. A117731 = Numerator of n*Sum[ 1/(n+k), {k, 1, n} ]. Cf. A082687 = Numerator of Sum[ 1/(n+k), {k, 1, n} ].

h=0; Do[ h=h+1/(n+1)/(2n+1); f=Numerator[n*h]; g=Numerator[h]; If[ !Equal[f,g], Print[ {n,f/g} ] ], {n,1,10000} ]

a(n) = A117731[ A125740(n) ] / A082687[ A125740(n) ].

A126563 Numbers k such that the ratio of A117731(k) and A082687(k) is composite.

Original entry on oeis.org

119, 735, 5145, 36015, 252105, 1764735, 12353145

Offset: 1

Alexander Adamchuk, Mar 12 2007, Jun 09 2007

a(1) = 7*17, a(2) = 3*5*7^2, a(3) = 3*5*7^3.
Corresponding composite terms in A125741 are {119, 49, 49, 49, 49, 49, 49, ...}.
A125741(n) is composite for n = {7, 16, 36, 91, 226, 510, 1131, ...}.

Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Cf. A082687, A117731, A125581, A125740, A125741, A126196, A126197.

h=0; Do[ h=h+1/(n+1)/(2n+1); f=Numerator[n*h]; g=Numerator[h]; If[ !Equal[f, g] && !PrimeQ[f/g], Print[ {n, f/g, FactorInteger[n], FactorInteger[f/g]} ] ], {n, 1, 10000} ]

f(n) = sum(k=1, n, 1/(n+k));
isok(k) = my(fk = f(k), q = numerator(k*fk)/numerator(fk)); (q!=1) && !isprime(q); \\ Michel Marcus, Mar 08 2023

Edited by Max Alekseyev, Jul 12 2019

a(5)-a(7) from Jinyuan Wang, Jul 10 2025

A007406 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^2.

Original entry on oeis.org

1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141, 1968329, 239437889, 240505109, 40799043101, 40931552621, 205234915681, 822968714749, 238357395880861, 238820721143261, 86364397717734821, 17299975731542641, 353562301485889, 354019312583809, 187497409728228241

Offset: 1

N. J. A. Sloane, Mira Bernstein

By Wolstenholme's theorem, p divides a(p-1) for prime p > 3. - T. D. Noe, Sep 05 2002
Also p divides a( (p-1)/2 ) for prime p > 3. - Alexander Adamchuk, Jun 07 2006
The rationals a(n)/A007407(n) converge to Zeta(2) = (Pi^2)/6 = 1.6449340668... (see the decimal expansion A013661).
For the rationals a(n)/A007407(n), n >= 1, see the W. Lang link under A103345 (case k=2).
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 squares. - Colin Barker, Nov 13 2014
Conjecture: for n > 3, gcd(n, a(n-1)) = A089026(n). Checked up to n = 10^5. - Amiram Eldar and Thomas Ordowski, Jul 28 2019
True if n is prime, by Wolstenholme's theorem. It remains to show that gcd(n, a(n-1)) = 1 if n > 3 is composite. - Jonathan Sondow, Jul 29 2019
From Peter Bala, Feb 16 2022: (Start)
Sum_{k = 1..n} 1/k^2 = 1 + (1 - 1/2^2)*(n-1)/(n+1) - (1/2^2 - 1/3^2)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/3^2 - 1/4^2)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - (1/4^2 - 1/5^2)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + .... Cf. A082687 and A120778.
This identity allows us to extend the definition of Sum_{k = 1..n} 1/k^2 to non-integral values of n. (End)
Numerators of the Eulerian numbers T(-2,k) for k = 0,1..., if T(n,k) is extended to negative n by the recurrence T(n,k) = (k+1)*T(n-1,k) + (n-k)*T(n-1,k-1) (indexed as in A173018). - Michael J. Collins, Oct 10 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..1152 (terms 1..200 from T. D. Noe)
Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012.
Romeo 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
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.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
Eric Weisstein's World of Mathematics, Wolstenholme Number

Cf. A001008, A007407 (denominators), A000290, A082687, A120778.

Numbers n such that a(n) is prime are listed in A111354. Primes in {a(n)} are listed in A123751. - Alexander Adamchuk, Oct 11 2006

import Data.Ratio ((%), numerator)
a007406 n = a007406_list !! (n-1)
a007406_list = map numerator $ scanl1 (+) $ map (1 %) $ tail a000290_list
-- Reinhard Zumkeller, Jul 06 2012

[Numerator(&+[1/k^2:k in [1..n]]):n in [1..23]]; // Marius A. Burtea, Aug 02 2019

a:= n-> numer(add(1/i^2, i=1..n)): seq(a(n), n=1..24);  # Zerinvary Lajos, Mar 28 2007

a[n_] := If[ n<1, 0, Numerator[HarmonicNumber[n, 2]]]; Table[a[n], {n, 100}]
Numerator[HarmonicNumber[Range[20],2]] (* Harvey P. Dale, Jul 06 2014 *)

{a(n) = if( n<1, 0, numerator( sum( k=1, n, 1 / k^2 ) ) )} /* Michael Somos, Jan 16 2011 */

Sum_{k=1..n} 1/k^2 = sqrt(Sum_{j=1..n} Sum_{i=1..n} 1/(i*j)^2). - Alexander Adamchuk, Oct 26 2004
G.f. for rationals a(n)/A007407(n), n >= 1: polylog(2,x)/(1-x).
a(n) = Numerator of (Pi^2)/6 - Zeta(2,n). - Artur Jasinski, Mar 03 2010

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

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

A120778 Numerators of partial sums of Catalan numbers scaled by powers of 1/4.

Original entry on oeis.org

1, 5, 11, 93, 193, 793, 1619, 26333, 53381, 215955, 436109, 3518265, 7088533, 28539857, 57414019, 1846943453, 3711565741, 14911085359, 29941580393, 240416274739, 482473579583, 1936010885087, 3883457090629, 62306843256889

Offset: 0

Wolfdieter Lang, Jul 20 2006

For denominators see A120777.
From the expansion of 0 = sqrt(1-1) = 1 - (1/2)*Sum_{k>=0} C(k)/4^k one has r:=lim_{n->infinity} r(n) = 2, with the partial sums r(n) defined below.
The series a(n)/A046161(n+1) is absolutely convergent to 1. - Ralf Steiner, Feb 16 2017
If n >= 1 it appears a(n-1) is equal to the difference between the denominator and the numerator of the ratio (2n)!!/(2n-1)!!. In particular a(n-1) is the difference between the denominator and the numerator of the ratio A001147(2n-1)/A000165(2n). See examples. - Anthony Hernandez, Feb 05 2020
From Peter Bala, Feb 16 2022: (Start)
Sum_{k = 0..n-1} Catalan(k)/4^k = (1/4^n)*(2*n)*binomial(2*n,n) *( 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. A082687 and A101028.
This identity allows us to extend the definition of Sum_{k = 0..n} Catalan(k)/4^k to non-integral values of n. (End)

			Rationals r(n): [1, 5/4, 11/8, 93/64, 193/128, 793/512, 1619/1024, 26333/16384, ...].
From _Anthony Hernandez_, Feb 05 2020: (Start)
For n = 4. The 4th even number is 8, and 8!!/(8-1)!! = 128/35, so a(4-1) = a(3) = 128 - 35 = 93.
For n = 7. The 7th even number is 14, and 14!!/(14-1)!! = 2048/429, so a(7-1) = a(6) = 2048 - 429 = 1619. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Wolfdieter Lang, Rationals r(n) and limit 2

Factors of A160481. Cf. A120777 (denominators), A082687, A101028, A141244.

[Numerator(2*(1-Binomial(2*n+2,n+1)/4^(n+1))): n in [0..25]]; // Vincenzo Librandi, Feb 17 2017

a := n -> 2^(2*(n+1) - add(i, i=convert(n+1, base, 2)))* (1-((n+1/2)!)/(sqrt(Pi)*(n+1)!)): seq(simplify(a(n)), n=0..23); # Peter Luschny, Dec 21 2017

f[n_] := f[n] = Numerator[(4/Pi) (n + 1) Integrate[x^n*ArcSin[Sqrt[x]], {x, 0, 1}]]; Array[f, 23, 0] (* Robert G. Wilson v, Jan 03 2011 *)
a[n_] := 2^(2(n+1) - DigitCount[n+1,2,1])(1 - ((n+1/2)!)/(Sqrt[Pi](n+1)!));
Table[a[n], {n, 0, 23}] (* Peter Luschny, Dec 21 2017 *)

a(n) = numerator(r(n)), with the rationals r(n):=Sum_{k = 0..n} C(k)/4^k with C(k) := A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
r(n) = (4/Pi)*(n+1)*Integral_{x = 0..1} x^n*arcsin(sqrt(x)) dx. - Groux Roland, Jan 03 2011
r(n) = 2*(1 - binomial(2*n+2,n+1)/4^(n+1)). - Groux Roland, Jan 04 2011
a(n) = A141244(2n+2) = A141244(2n+3) (conjectural). - Greg Martin, Aug 16 2014, corrected by M. F. Hasler, Aug 18 2014
From Peter Luschny, Dec 21 2017: (Start)
a(n) = numerator(1 - ((n+1/2)!)/(sqrt(Pi)*(n+1)!)).
a(n) = 2^(2*(n+1) - HammingWeight(n+1))*(1 - ((n+1/2)!)/(sqrt(Pi)*(n+1)!)). (End)

A101028 Numerator of partial sums of a certain series. First member (m = 2) of a family.

Original entry on oeis.org

1, 11, 79, 479, 5297, 69071, 69203, 471181, 8960447, 44831407, 1031626241, 5160071143, 15484789693, 64166447971, 1989542332021, 3979714828967, 27861681000449, 1030996803010973, 1031094241305773, 42278288849598913, 1818093633186492859, 1818204269645957299, 85460151199040573933

Offset: 1

Wolfdieter Lang, Dec 17 2004

The denominators are given in A101029.
The limit s = lim_{n -> infinity} s(n) with s(n) defined below equals 3*Sum_{k >= 1} zeta(2*k+1)/2^(2*k) with Euler's (or Riemann's) zeta function. This limit is 3*(2*log(2)-1) = 1.158883083...; see the Abramowitz-Stegun reference p. 259, eq. 6.3.15 with z = 1/2 together with p. 258, eqs. 6.3.5 and 6.3.3.
This is the first member (m = 2) of a family of rational partial sum sequences s(n,m) = (m-1)*m*(m+1)*Sum_{k = 1..n} 1/((m*k-1)*(m*k)*(m*k+1)) which have limit s(m) = lim_{n -> infinity} s(n,m) = -(gamma + Psi(1/m) + m/2 + Pi*cot(Pi*x)/2), with the Euler-Mascheroni constant gamma and the digamma function Psi. The same limit is reached by (m^2-1)*Sum_{k >= 0} zeta(2*k+1)/m^(2*k).
From Peter Bala, Feb 17 2022: (Start)
Let F(n) = (6*n/(2*n-1))*( 1/(1*2)*(n-1)/n - 1/(2*3)*(n-1)*(n-2)/(n*(n+1)) + 1/(3*4)*(n-1)*(n-2)*(n-3)/(n*(n+1)*(n+2)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/(n*(n+1)*(n+2)*(n+3)) + ...). Then F(n+1) = 6*Sum_{k = 1..n} 1/((2*k-1)*(2*k)*(2*k+1)). Cf. A082687.
This identity allows us to extend the definition of Sum_{k = 1..n} 1/((2*k-1)*(2*k)*(2*k+1)) to non-integral values of n. (End)

			s(3)= 6*(1/(1*2*3)+ 1/(3*4*5) + 1/(5*6*7)) = 79/70, hence a(3)=79 and A101029(3)=70.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 258-259.
Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

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, Rationals s(n) and more.

Cf. A101627 (m=3), A101629 (m=4), A101631 (m=5).

Cf. A082687, A101029 (denominators).

a(n) = numerator(6*sum(k=1, n, 1/((2*k-1)*(2*k)*(2*k+1)))); \\ Michel Marcus, Feb 28 2022

a(n) = numerator(s(n)) with s(n) = 6*Sum_{k = 1..n} 1/((2*k-1)*(2*k)*(2*k+1)).

More terms from Michel Marcus, Feb 28 2022

A117664 Denominator of the sum of all elements in the n X n Hilbert matrix M(i,j) = 1/(i+j-1), where i,j = 1..n.

Original entry on oeis.org

1, 3, 10, 105, 252, 2310, 25740, 9009, 136136, 11639628, 10581480, 223092870, 1029659400, 2868336900, 11090902680, 644658718275, 606737617200, 4011209802600, 140603459396400, 133573286426580, 5215718803323600

Offset: 1

Alexander Adamchuk, Apr 11 2006

nonn

Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1) = A117731(n) / A117664(n) = 2n * H'(2n) = 2n * A058313(2n) / A058312(2n), where H'(2n) is 2n-th alternating sign Harmonic Number. H'(2n) = H(2n) - H(n), where H(n) is n-th Harmonic Number. - Alexander Adamchuk, Apr 23 2006

			For n=2, the 2 X 2 Hilbert matrix is [1, 1/2; 1/2, 1/3], so a(2) = denominator(1 + 1/2 + 1/2 + 1/3) = denominator(7/3) = 3.
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 ...
  ...

Eric Weisstein's World of Mathematics, Hilbert Matrix.
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A091342, A098118, A111876, A082687, A086881, A005249, A001008, A002805.

Numerator is A117731(n).

Table[Denominator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 30}]

a(n) = A111876(n-1)/n.
a(n) = denominator( Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1) ). Numerator is A117731(n). - Alexander Adamchuk, Apr 23 2006
a(n) = denominator( Sum_{k=1..n} (2*k)/(n+k) ). - Peter Bala, Oct 10 2021

A111876 Denominator of Sum_{k = 0..n} 1/((k+1)(2k+1)).

Original entry on oeis.org

1, 6, 30, 420, 1260, 13860, 180180, 72072, 1225224, 116396280, 116396280, 2677114440, 13385572200, 5736673800, 166363540200, 10314539492400, 10314539492400, 72201776446800, 2671465728531600, 2671465728531600

Offset: 0

Paul Barry, Aug 19 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A082687 (numerators), A117664.

[Denominator(HarmonicNumber(2*n+2) -HarmonicNumber(n+1))/2: n in [0..40]]; // G. C. Greubel, Jul 24 2023

seq(denom( add(1/((k+1)*(2*k+1)), k = 0..n) ), n = 0..20); # Peter Bala, Oct 10 2021

Table[Denominator[HarmonicNumber[2n+2] - HarmonicNumber[n+1]]/2, {n, 0, 30}]

a(n) = denominator(sum(k=0, n, 1/((k+1)*(2*k+1)))); \\ Michel Marcus, Oct 10 2021

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

a(n) = denominator of ( digamma(n+3/2) - digamma(n+2) + 2*log(2) ).
a(n) = denominator of 2*(n+1)*Integral_{x = 0..1} x^n* log(1+sqrt(x)) dx.
a(n-1) = denominator( (1/n)*Sum_{k = 1..n} (n - k)/(n + k) ). - Peter Bala, Oct 10 2021

cp's OEIS Frontend

A125740 Numbers n such that A117731(n) differs from A082687(n).

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A125741 The ratio of A117731(n) and A082687(n) when they are different.

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A126563 Numbers k such that the ratio of A117731(k) and A082687(k) is composite.

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A007406 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^2.

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

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A120778 Numerators of partial sums of Catalan numbers scaled by powers of 1/4.

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A111876 Denominator of Sum_{k = 0..n} 1/((k+1)(2k+1)).