The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A064169 Numerator - denominator in n-th harmonic number, 1 + 1/2 + 1/3 + ... + 1/n. 10
 0, 1, 5, 13, 77, 29, 223, 481, 4609, 4861, 55991, 58301, 785633, 811373, 835397, 1715839, 29889983, 10190221, 197698279, 40315631, 13684885, 13920029, 325333835, 990874363, 25128807667, 25472027467, 232222818803, 235091155703, 6897956948587, 6975593267347 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The numerator and denominator in the definition have no common factors greater than 1. p divides a(p-2) for prime p > 2. - Alexander Adamchuk, Jun 09 2006 It appears that a(n) = numerator((3*(HarmonicNumber(n) - 1)) / (n*(n^2 + 6*n + 11)), except for n = 5, 82, 115, and 383 (tested to 20000). - Gary Detlefs, Jul 20 2011 From Amiram Eldar and Thomas Ordowski, Jul 27 2019: (Start) Conjecture: for n > 2, n divides a(n-2) if and only if n is a prime. Checked up to 20000. Max Alekseyev proved (in priv. commun.) that there are no primes p > 3 such that p^2 divides a(p-2). (End) LINKS Harvey P. Dale, Table of n, a(n) for n = 1..1000 Eric Weisstein's World of Mathematics, Harmonic Number FORMULA Numerator of (gamma + Psi(n+1) - 1). - Vladeta Jovovic, Aug 12 2002 From Alexander Adamchuk, Jun 09 2006: (Start) a(n) = numerator of Sum_{k = 2..n} 1/k. a(n) = A001008(n) - A002805(n). a(n) = numerator of (the n-th harmonic number minus 1). a(n) = numerator of A001008(n)/A002805(n) - 1. (End) a(n) = numerator of A027612(n-1)/(A027611(n)*n^2*(n-1)!), n > 1. - Gary Detlefs, Aug 05 2011 a(n) = numerator(Sum_{k = 1..n-1} 1/(3*k + 3)). - Gary Detlefs, Sep 14 2011 a(n) = numerator(Sum_{k = 0..n-1} 2/(k+2)). - Gary Detlefs, Oct 06 2011 a(n) = numerator(Sum_{k = 1..n} frac(1/k)). - Michel Marcus, Sep 27 2021 EXAMPLE The 3rd harmonic number is 11/6. So a(3) = 11 - 6 = 5. MAPLE s := n -> add(1/i, i=2..n): a := n -> numer(s(n)): seq(a(n), n=1..30); # Zerinvary Lajos, Mar 28 2007 MATHEMATICA A064169[n_]:= (s = Sum[1/k, {k, n}]; Numerator[s] - Denominator[s]); Table[A064169[n], {n, 35}] Numerator[Table[Sum[1/k, {k, 2, n}], {n, 35}]] (* Alexander Adamchuk, Jun 09 2006 *) Numerator[#] - Denominator[#] &/@ HarmonicNumber[Range[35]] (* Harvey P. Dale, Apr 25 2016 *) Numerator[Accumulate[1/Range[2, 35]]] (* Alonso del Arte, Nov 21 2018 *) a[n_] := Numerator[PolyGamma[1 + n] + EulerGamma - 1]; Table[a[n], {n, 1, 29}] (* Peter Luschny, Feb 19 2022 *) PROG (PARI) a(n) = my(h=sum(i=1, n, 1/i)); numerator(h)-denominator(h) \\ Felix Fröhlich, Jan 14 2019 (Magma) [Numerator(a)-Denominator(a) where a is HarmonicNumber(n): n in [1..35]]; // Marius A. Burtea, Aug 03 2019 (Sage) [numerator(harmonic_number(n)) - denominator(harmonic_number(n)) for n in (1..35)] # G. C. Greubel, Jul 27 2019 (GAP) List([1..35], n-> NumeratorRat(Sum([0..n-2], k-> 2/(k+2))) ); # G. C. Greubel, Jul 27 2019 (Python) from sympy import harmonic def A064169(n): return (lambda x: x.p - x.q)(harmonic(n)) # Chai Wah Wu, Sep 27 2021 CROSSREFS Cf. A001008, A002805, A064167, A064168. Sequence in context: A163732 A208821 A293259 * A294208 A081525 A027612 Adjacent sequences: A064166 A064167 A064168 * A064170 A064171 A064172 KEYWORD nonn AUTHOR Leroy Quet, Sep 19 2001 EXTENSIONS One more term from Robert G. Wilson v, Sep 28 2001 More terms from Vladeta Jovovic, Aug 12 2002 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 12 13:01 EDT 2024. Contains 374247 sequences. (Running on oeis4.)