A027611 Denominator of n * n-th harmonic number.

1, 1, 2, 3, 12, 10, 20, 35, 280, 252, 2520, 2310, 27720, 25740, 24024, 45045, 720720, 680680, 4084080, 3879876, 739024, 235144, 5173168, 14872858, 356948592, 343219800, 2974571600, 2868336900, 80313433200, 77636318760

Offset: 1

Author

Glen Burch (gburch(AT)erols.com)

Keywords

• nonn
• easy
• frac

Comments

This is very similar to A128438, which is a different sequence. They differ at n=6 (and nowhere else?). - N. J. A. Sloane, Nov 21 2008

Denominator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n.

Denominator of Sum_{k=1..n} frac(n/k) where frac(x/y) denotes the fractional part of x/y. - Benoit Cloitre, Oct 03 2002

Denominator of Sum_{d=2..n-1, n mod d > 0} n/d. Numerator = A079076. - Reinhard Zumkeller, Dec 21 2002

a(n) is odd iff n is a power of 2. - Benoit Cloitre, Oct 03 2002

Indices where a(n) differs from A128438 are terms of A074791. - Gary Detlefs, Sep 03 2011

Links

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

Crossrefs

Harmonic numbers = A001008/A002805.

Cf. A001705, A006675, A027612, A049820, A024816.

Cf. A128438, A074791, A079076.

Programs

Haskell

import Data.Ratio ((%), denominator)
a027611 n = denominator $ sum $ map (n %) [1..n]
-- Reinhard Zumkeller, Jul 03 2012

Magma

[Denominator(n*HarmonicNumber(n)): n in [1..40]]; // Vincenzo Librandi, Feb 19 2014

Maple

a := n -> denom(add((n-j)/j, j=1..n));
seq(a(n), n = 1..30); # Peter Luschny, May 12 2023

Mathematica

a[n_]:=Denominator[n*HarmonicNumber[n]]; Array[a,100] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011 *)

PARI

a(n) = denominator(n*sum(k=1, n, 1/k)); \\ Michel Marcus, Feb 15 2015

Python

from sympy import harmonic
def A027611(n): return (n*harmonic(n)).q # Chai Wah Wu, Sep 26 2021

SageMath

[denominator(n*harmonic_number(n)) for n in (1..40)] # G. C. Greubel, Aug 24 2022

Formula

From Vladeta Jovovic, Sep 02 2002: (Start)

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

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

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

a(n) = numerator(h(n)/h(n-1)) - denominator(h(n)/h(n-1)), n > 1, where h(n) is the n-th harmonic number. - Gary Detlefs, Sep 03 2011

a(n) = A213999(n, n-2) for n > 1. - Reinhard Zumkeller, Jul 03 2012

a(n) = denominators of coefficients of e.g.f. -1 + exp(x)*(1 + Sum_{j >= 0} (-x)^(j+1)/(j * j!)). - G. C. Greubel, Aug 24 2022

Extensions

Entry revised by N. J. A. Sloane following a suggestion of Eric W. Weisstein, Jul 02 2004