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

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

Offset: 1

Author

Benoit Cloitre, Apr 12 2003

Keywords

• frac
• nonn
• easy

Comments

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)

Examples

			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 ...
		

Links

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

Crossrefs

Bisection of A058313, A082688 (denominators).

Cf. A001008, A002805, A005249, A007406, A058312.

Cf. A086881, A098118, A101028, A117731, A120778.

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

SageMath

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

Formula

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)).