A117664 -id:A117664 - cp's OEIS frontend

A119030 Difference between numerator and denominator of the sum of all matrix elements of n X n Hilbert matrix M(i,j)=1/(i+j-1) (i,j = 1..n), A117731[n] - A117664[n].

0, 4, 27, 428, 1375, 15797, 211631, 86540, 1496205, 144045379, 145607407, 3378951221, 17021747431, 51392118293, 214084856611, 13337033800292, 13393340889767, 94103945740529, 3493457389196573, 3503912518228613

Offset: 1

Alexander Adamchuk, Jul 22 2006

nonn

p^3 divides a(p^k) for prime p>2 and integer k>0.

Cf. A117731, A117664.

Numerator[Table[Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}],{n,1,25}]] - Denominator[Table[Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}],{n,1,25}]]

a(n) = Numerator[Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}]] - Denominator[Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}]]. a(n) = A117731[n] - A117664[n].

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

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

A296519 Denominator of n*Sum_{k=1..n} 1/(n+k).

Original entry on oeis.org

2, 6, 20, 210, 504, 4620, 51480, 18018, 272272, 23279256, 21162960, 446185740, 2059318800, 5736673800, 22181805360, 1289317436550, 1213475234400, 8022419605200, 281206918792800, 267146572853160, 10431437606647200, 428163098127382800, 409547311252279200

Offset: 1

Eric W. Weisstein, Dec 14 2017

a(n) is divisible by all primes p such that the numerator of Sum_{n < k*p <= n} 1/k is not divisible by p, in particular by all primes from n+1 to 2*n-1. - Robert Israel, May 21 2020

			The first few fractions are 1/2, 7/6, 37/20, 533/210, 1627/504, 18107/4620, 237371/51480, ... = A117731/a(n).

Robert Israel, Table of n, a(n) for n = 1..1155

Cf. A111876, A117731 (numerators), A117664.

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

N:= 30: # for a(1)..a(N)
H:= ListTools:-PartialSums([seq(1/i,i=1..2*N)]):
map(n -> denom(n*(H[2*n]-H[n])), [$1..N]); # Robert Israel, May 21 2020

Table[n (HarmonicNumber[2 n] - HarmonicNumber[n]), {n, 30}] // Denominator

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

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

From G. C. Greubel, Jul 24 2023: (Start)
a(n) = 2*A117664(n).
a(n) = 2*A111876(n-1)/n. (End)

cp's OEIS Frontend

A119030 Difference between numerator and denominator of the sum of all matrix elements of n X n Hilbert matrix M(i,j)=1/(i+j-1) (i,j = 1..n), A117731[n] - A117664[n].

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

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

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A296519 Denominator of n*Sum_{k=1..n} 1/(n+k).

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Magma

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