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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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

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

EXAMPLE

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

MAPLE

A058313 := n->denom(add((-1)^(k+1)/k, k=1..n));

# Alternative:

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 *)

PROG

(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

(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

CROSSREFS

Numerators are A058313. Cf. A025530.