A064169 Numerator - denominator in n-th harmonic number, 1 + 1/2 + 1/3 + ... + 1/n.

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

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