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A064169 Numerator - denominator in n-th harmonic number, 1 + 1/2 + 1/3 +...+ 1/n. 4
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 > 1.

Numerator of ( HarmonicNumber[n] - 1 ). - Alexander Adamchuk, Jun 09 2006

p divides a(p-2) for prime p > 3. - 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 20,000). - Gary Detlefs, Jul 20 2011

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Harmonic Number

FORMULA

Numerators of gamma+Psi(n+1)-1. - Vladeta Jovovic, Aug 12 2002

From Alexander Adamchuk, Jun 09 2006: (Start)

a(n) = numerator(Sum_{k=2..n} 1/k).

a(n) = A001008(n) - A002805(n).

a(n) = numerator(HarmonicNumber(n) - 1).

a(n) = numerator(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

EXAMPLE

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

MAPLE

ZL:=n->sum(1/i, i=2..n): a:=n->floor(numer(ZL(n))): seq(a(n), n=1..28); # Zerinvary Lajos, Mar 28 2007

MATHEMATICA

f[ n_ ] := (s = Sum[ 1/k, {k, 1, n} ]; Numerator[ s ] - Denominator[ s ]); Table[ f[ n ], {n, 1, 25} ]

Numerator[Table[Sum[1/k, {k, 2, n}], {n, 1, 30}]] (* Alexander Adamchuk, Jun 09 2006 *)

Numerator[#]-Denominator[#]&/@HarmonicNumber[Range[30]] (* Harvey P. Dale, Apr 25 2016 *)

CROSSREFS

Cf. A001008, A002805.

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

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Last modified January 22 16:36 EST 2018. Contains 298055 sequences.