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

1,3

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

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

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 *)

Prog

(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
(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

Crossrefs

Harmonic numbers = A001008/A002805.

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

Cf. A128438, A074791, A079076.

Sequence in context: A081526 A075711 A079077 * A303221 A345049 A168059

Adjacent sequences: A027608 A027609 A027610 * A027612 A027613 A027614

Keyword

nonn,easy,frac

Author

Glen Burch (gburch(AT)erols.com)

Extensions

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

Status

approved