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%I #90 Mar 06 2024 15:57:22
%S 1,1,5,7,47,37,319,533,1879,1627,20417,18107,263111,237371,52279,
%T 95549,1768477,1632341,33464927,155685007,166770367,156188887,
%U 3825136961,3602044091,19081066231,18051406831,57128792093,7751493599,236266661971
%N Numerator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.
%C 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
%C For the limit n -> infinity of the partial sums of the alternating harmonic series see A002162. - _Wolfdieter Lang_, Sep 08 2015
%C a(n)/A058312(n) appears in the locker puzzle (see the links in A364317) as the probability of failures with the strategy used for n lockers and opening of up to floor(n/2) lockers. Note the alternative formula given below for a(n)/A058312(n) using only positive fractions. - _Wolfdieter Lang_, Aug 12 2023
%D L. B. W. Jolley, Summation of Series, Dover Publications, 1961, page 14, #71.
%H T. D. Noe and Robert Israel, <a href="/A058313/b058313.txt">Table of n, a(n) for n = 1..2000</a> (terms up to a(200) from T. D. Noe)
%H Khristo N. Boyadzhiev, <a href="https://sciendo.com/pdf/10.2478/tmmp-2013-0031">Power series with skew-harmonic numbers, dilogarithms, and double integrals</a>, Tatra Mt. Math. Publ., Vol. 56 (2013), pp. 93-108.
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/log2_000.htm">Factorizations of many number sequences</a>.
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/log2_100.htm">Factorizations of many number sequences</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%F G.f. for a(n)/A058312(n): log(1+x)/(1-x). - _Benoit Cloitre_, Jun 15 2003
%F 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
%F From _Peter Luschny_, May 03 2016: (Start)
%F 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:
%F AH(n) = H(n) - H((n - n mod 2)/2).
%F AH(z) = log(2)+(1/2)*cos(Pi*z)*(Psi(z/2+1/2)-Psi(z/2+1)).
%F AH(z) ~ log(2)+(1/2)*cos(Pi*z)*(-1/z+1/(2*z^2)-1/(4*z^4)+1/(2*z^6)-...).
%F AH(z) ~ log(2)-(1/2)*cos(Pi*z)*Sum_{n>=0} Euler(n,0)/z^(n+1). (End)
%F Sum_{k>=1} (-1)^(k+1)*AH(k)/k = Pi^2/12 + log(2)^2/2 (Boyadzhiev, 2013). - _Amiram Eldar_, Oct 04 2021
%F a(n)/A058312(n) = Sum_{j=0..ceiling(n/2) - 1} 1/(n-j), for n >= 1. (Proof by comparing the recurrences for even and odd n.) - _Wolfdieter Lang_, Aug 12 2023
%F For n >= 1, log(2) = a(n)/A058312(n) + (-1)^n*n!*Sum_{k >= 1} 1/(k*(k + 1)* ...*(k + n)*2^k). - _Peter Bala_, Dec 07 2023
%F a(n) = the (reduced) numerator of the continued fraction 1/(1 + 1^2/(1 + 2^2/(1 + 3^2/(1 + ... + (n-1)^2/(1))))). - _Peter Bala_, Feb 18 2024
%e 1, 1/2, 5/6, 7/12, 47/60, 37/60, 319/420, 533/840, 1879/2520, ...
%e For n=4: a(n)/A058312(n) = 7/12 because 1/1 - 1/2 + 1/3 - 1/4 = 7/12 = 1/4 + 1/3. - _Wolfdieter Lang_, Aug 12 2023
%p A058313 := n->numer(add((-1)^(k+1)/k,k=1..n));
%p # Alternatively:
%p a := n -> numer(harmonic(n) - harmonic((n-modp(n,2))/2)):
%p seq(a(n), n=1..29); # _Peter Luschny_, May 03 2016
%t Numerator[Table[Sum[(-1)^(k + 1)/k, {k, n}], {n, 30}]] (* _Harvey P. Dale_, Jul 18 2012 *)
%t a[n_]:= (-1)^n (HarmonicNumber[n/2 - 1/2] - HarmonicNumber[n/2] + (-1)^n Log[4])/2; Table[a[n] // FullSimplify, {n, 29}] // Numerator (* _Gerry Martens_, Jul 05 2015 *)
%t Rest[Numerator[CoefficientList[Series[Log[1 + x]/(1 - x), {x, 0, 33}], x]]] (* _Vincenzo Librandi_, Jul 06 2015 *)
%t Table[Log[2] - (-1)^n LerchPhi[-1, 1, n + 1], {n, 20}] // Numerator (* _Eric W. Weisstein_, Aug 25 2023 *)
%o (PARI) a(n)=(-1)^n*numerator(polcoeff(log(1-x)/(x+1)+O(x^(n+1)), n))
%o (Haskell)
%o import Data.Ratio((%), numerator)
%o a058313 n = a058313_list !! (n-1)
%o a058313_list = map numerator $ scanl1 (+) $ map (1 %) $ tail a181983_list
%o -- _Reinhard Zumkeller_, Mar 20 2013
%Y Denominators are A058312. Cf. A025530.
%Y Apart from leading term, same as A075830.
%Y Cf. A001008 (numerator of n-th harmonic number).
%Y Bisections are A049281 and A082687.
%Y Cf. A181983.
%K nonn,frac,nice,easy
%O 1,3
%A _N. J. A. Sloane_, Dec 09 2000
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