A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n.

1, 5, 13, 77, 87, 223, 481, 4609, 4861, 55991, 58301, 785633, 811373, 835397, 1715839, 29889983, 30570663, 197698279, 201578155, 41054655, 13920029, 325333835, 990874363, 25128807667, 25472027467, 232222818803, 235091155703, 6897956948587, 6975593267347

Offset: 1

Author

Glen Burch (gburch(AT)erols.com)

Keywords

• nonn
• easy
• frac

Comments

Numerator of a second-order harmonic number H(n, (2)) = Sum_{k=1..n} HarmonicNumber(k). - Alexander Adamchuk, Apr 12 2006

p divides a(p-3) for prime p > 3. - Alexander Adamchuk, Jul 06 2006

Denominator is A027611(n+1). p divides a(p-3) for prime p > 3. - Alexander Adamchuk, Jul 26 2006

a(n) = A213998(n,n-2) for n > 1. - Reinhard Zumkeller, Jul 03 2012

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Harmonic Number.

Crossrefs

Cf. A001008, A001705, A002805, A006675, A022819, A027611, A093418.

Programs

Haskell

import Data.Ratio ((%), numerator)
a027612 n = numerator $ sum $ zipWith (%) [1 .. n] [n, n-1 .. 1]
-- Reinhard Zumkeller, Jul 03 2012

Magma

[Numerator((&+[j/(n-j+1): j in [1..n]])): n in [1..30]]; // G. C. Greubel, Aug 23 2022

Maple

a := n -> numer(add((n+1-j)/j, j=1..n));
seq(a(n), n = 1..29); # Peter Luschny, May 12 2023

Mathematica

Numerator[Table[Sum[Sum[1/i,{i,1,k}],{k,1,n}],{n,1,30}]] (* Alexander Adamchuk, Apr 12 2006 *)
Numerator[Table[Sum[k/(n-k+1),{k,1,n}],{n,1,50}]] (* Alexander Adamchuk, Jul 26 2006 *)

PARI

a(n) = numerator(sum(k=1, n, k/(n-k+1))); \\ Michel Marcus, Jul 14 2018

SageMath

[numerator(n*(harmonic_number(n+1) - 1)) for n in (1..30)] # G. C. Greubel, Aug 23 2022

Formula

From Vladeta Jovovic, Sep 02 2002: (Start)

a(n) = numerators of coefficients in expansion of -log(1-x)/(1-x)^2.

a(n) = numerators of (n+1)*(harmonic(n+1) - 1).

a(n) = numerators of (n+1)*(Psi(n+2) + Euler-gamma - 1). (End)

a(n) = numerator( Sum_{k=1..n} Sum_{i=1..k} 1/i ). - Alexander Adamchuk, Apr 12 2006

a(n) = numerator( Sum_{k=1..n} k/(n-k+1) ). - Alexander Adamchuk, Jul 26 2006

a(n) = numerator of integral_{x=1..n+1} floor((n+1)/x). - Jean-François Alcover, Jun 18 2013