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

Author

Leroy Quet, Sep 19 2001

Keywords

• nonn

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)

Examples

			The 3rd harmonic number is 11/6. So a(3) = 11 - 6 = 5.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

Cf. A001008, A002805, A064167, A064168.

Programs

GAP

List([1..35], n-> NumeratorRat(Sum([0..n-2], k-> 2/(k+2))) ); # G. C. Greubel, Jul 27 2019

Magma

[Numerator(a)-Denominator(a) where a is HarmonicNumber(n): n in [1..35]]; // Marius A. Burtea, Aug 03 2019

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 *)

PARI

a(n) = my(h=sum(i=1, n, 1/i)); numerator(h)-denominator(h) \\ Felix Fröhlich, Jan 14 2019

Python

from sympy import harmonic
def A064169(n): return (lambda x: x.p - x.q)(harmonic(n)) # Chai Wah Wu, Sep 27 2021

Sage

[numerator(harmonic_number(n)) - denominator(harmonic_number(n)) for n in (1..35)] # G. C. Greubel, Jul 27 2019

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

Extensions

One more term from Robert G. Wilson v, Sep 28 2001

More terms from Vladeta Jovovic, Aug 12 2002