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A058313 Numerator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k. 46
1, 1, 5, 7, 47, 37, 319, 533, 1879, 1627, 20417, 18107, 263111, 237371, 52279, 95549, 1768477, 1632341, 33464927, 155685007, 166770367, 156188887, 3825136961, 3602044091, 19081066231, 18051406831, 57128792093, 7751493599, 236266661971 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A Wolstenholme-like theorem: for prime p > 3, if p = 6k-1, then p divides a(4k-1), otherwise if p = 6k+1, then p divides a(4k). - T. D. Noe, Apr 01 2004

For the limit n -> infinity of the partial sums of the alternating harmonic series see A002162. - Wolfdieter Lang, Sep 08 2015

LINKS

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..2000 (terms up to a(200) from T. D. Noe)

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

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) + (-1)^(n+1)*A058312(n-1))/gcd(n*a(n-1) + (-1)^(n+1)*A058312(n-1), n*A058312(n-1)). - Robert Israel, Jul 06 2015

From Peter Luschny, May 03 2016: (Start)

Let H(n) denote the harmonic numbers, AH(n) denote the alternating harmonic numbers, Psi the polygamma function and euler(n,x) the Euler polynomials. Then:

AH(n) = H(n) - H((n - n mod 2)/2).

AH(z) = log(2)+(1/2)*cos(Pi*z)*(Psi(z/2+1/2)-Psi(z/2+1)).

AH(z) ~ log(2)+(1/2)*cos(Pi*z)*(-1/z+1/(2*z^2)-1/(4*z^4)+1/(2*z^6)-...).

AH(z) ~ log(2)-(1/2)*cos(Pi*z)*Sum_{n>=0} euler(n,0)/z^(n+1). (End)

EXAMPLE

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

MAPLE

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

# Alternatively:

a := n -> numer(harmonic(n) - harmonic((n-modp(n, 2))/2)):

seq(a(n), n=1..29); # Peter Luschny, May 03 2016

MATHEMATICA

Numerator[Table[Sum[(-1)^(k+1)/k, {k, n}], {n, 30}]] (* Harvey P. Dale, Jul 18 2012 *)

a[n_]:= (-1)^n(HarmonicNumber[n/2-1/2]-HarmonicNumber[n/2]+(-1)^n Log[4])/2; Table[a[n] // FullSimplify, {n, 1, 29}] // Numerator (* Gerry Martens, Jul 05 2015 *)

Rest[Numerator[CoefficientList[Series[Log[1 + x]/(1 - x), {x, 0, 33}], x]]] (* Vincenzo Librandi, Jul 06 2015 *)

PROG

(PARI) a(n)=(-1)^n*numerator(polcoeff(log(1-x)/(x+1)+O(x^(n+1)), n))

(Haskell)

import Data.Ratio((%), numerator)

a058313 n = a058313_list !! (n-1)

a058313_list = map numerator $ scanl1 (+) $ map (1 %) $ tail a181983_list

-- Reinhard Zumkeller, Mar 20 2013

CROSSREFS

Denominators are A058312. Cf. A025530.

Apart from leading term, same as A075830.

Cf. A001008 (numerator of n-th harmonic number).

Bisections are A049281 and A082687.

Cf. A181983.

Sequence in context: A174267 A306649 A075830 * A120301 A119787 A025530

Adjacent sequences:  A058310 A058311 A058312 * A058314 A058315 A058316

KEYWORD

nonn,frac,nice,easy

AUTHOR

N. J. A. Sloane, Dec 09 2000

STATUS

approved

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Last modified May 19 17:48 EDT 2019. Contains 323395 sequences. (Running on oeis4.)