A102928 Numerator of the harmonic mean of the first n positive integers.

1, 4, 18, 48, 300, 120, 980, 2240, 22680, 25200, 304920, 332640, 4684680, 5045040, 5405400, 11531520, 208288080, 73513440, 1474352880, 62078016, 108636528, 113809696, 2736605872, 8566766208, 223092870000, 232016584800

Offset: 1

Author

Eric W. Weisstein, Jan 19 2005

Keywords

• nonn
• frac

Comments

See A175441 - denominators of the harmonic means of the first n positive integers. - Jaroslav Krizek, May 16 2010

a(n) is also the denominator of H(n-1)/n + 1/n^2 = -Integral_{x=0..1} x^n*log(1-x) with H(n) = A001008(n)/A002805(n) the harmonic number of order n. - Groux Roland, Jan 08 2011 [corrected by Gary Detlefs, Oct 06 2011]

Equivalently, a(n) is the reduced denominator of the arithmetic mean of the reciprocals of the first n positive integers (corresponding reduced numerator is A175441(n)). - Rick L. Shepherd, Jun 15 2014

n divides a(n) iff n is not from the sequence A256102. - Wolfdieter Lang, Apr 23 2015

Examples

			1, 4/3, 18/11, 48/25, 300/137, 120/49, 980/363, 2240/761, ...
Division property: The first n not dividing a(n) is 20 because 20 = A256102(1). Indeed, a(20) = 62078016. - _Wolfdieter Lang_, Apr 23 2015
		

Links

• Stefano Spezia, Table of n, a(n) for n = 1..2000
• Eric Weisstein's World of Mathematics, Harmonic Mean

Crossrefs

Cf. A001008, A175441, A256102.

Programs

Mathematica

Table[Numerator[n/HarmonicNumber[n]], {n, 26}]

PARI

a(n) = numerator(n/sum(k=1, n, 1/k)); \\ Michel Marcus, Jul 29 2022

Formula

a(n) = denominator(EulerGamma/n + PolyGamma(0, 1 + n)/n). - Artur Jasinski, Nov 02 2008

a(n) = numerator(n/H(n)), where H(n) is the n-th harmonic number. - Gary Detlefs, Sep 10 2011

a(n) = denominator((1/n)*Sum_{k=1..n} k + 1/k). - Stefano Spezia, Jul 27 2022

a(n) = denominator(Sum_{k>0} 1/(k*(k+n))). - Mohammed Yaseen, Jun 23 2024

Extensions

Definition edited by N. J. A. Sloane, Jan 24 2024