A058312 Denominator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.

1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360, 360360, 72072, 144144, 2450448, 2450448, 46558512, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 11473347600

Offset: 1

Author

N. J. A. Sloane, Dec 09 2000

Keywords

• nonn
• frac
• nice
• easy

Comments

a(n) is a divisor of A003418(n). The first time this is a proper divisor, is a(15); see A269626. - Jeppe Stig Nielsen, Mar 01 2016

Examples

			1, 1/2, 5/6, 7/12, 47/60, 37/60, 319/420, 533/840, 1879/2520, ...
		

Links

• T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..2000 (1..200 from T. D. Noe)
• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

Numerators are A058313. Cf. A025530.

Cf. A002805 (denominator of n-th harmonic number).

Cf. A121594, A181983, A003418, A269626.

Programs

Haskell

import Data.Ratio((%), denominator)
a058312 n = a058312_list !! (n-1)
a058312_list = map denominator $ scanl1 (+) $
                   map (1 %) $ tail a181983_list
-- Reinhard Zumkeller, Mar 20 2013

Maple

A058313 := n->denom(add((-1)^(k+1)/k,k=1..n));
# Alternatively:
a := n -> denom(harmonic(n) - harmonic((n-modp(n,2))/2)):
seq(a(n), n=1..28); # Peter Luschny, May 03 2016

Mathematica

a[n_] := Sum[(-1)^(k+1)/k, {k, 1, n}]; Table[a[n] // Denominator, {n, 1, 30}] (* Jean-François Alcover, May 26 2015 *)
a[n_]:= (-1)^n(HarmonicNumber[n/2-1/2]-HarmonicNumber[n/2]+(-1)^n Log[4])/2; Table[a[n] // FullSimplify, {n, 1, 29}] // Denominator (* Gerry Martens, Jul 05 2015 *)
Rest[Denominator[CoefficientList[Series[Log[1 + x]/(1 - x),{x, 0, 33}], x]]] (* Vincenzo Librandi, Jul 06 2015 *)

PARI

a(n)=denominator(polcoeff(-log(1-x)/(x+1)+O(x^(n+1)),n))

PARI

a(n)=denominator(sum(k=1,n,(-1)^(k+1)/k)) \\ Jeppe Stig Nielsen, Mar 01 2016

Formula

G.f. for A058313(n)/ A058312(n): log(1+x)/(1-x). - Benoit Cloitre, Jun 15 2003

a(n) = n*a(n-1)/gcd(n*a(n-1), n*A058313(n-1)+(-1)^(n-1)*a(n-1)). - Robert Israel, Jul 05 2015

a(n) = the (reduced) denominator of the continued fraction 1/(1 + 1^2/(1 + 2^2/(1 + 3^2/(1 + ... + (n-1)^2/(1))))). - Peter Bala, Feb 18 2024