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

1,2

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

Links

Stefano Spezia, Table of n, a(n) for n = 1..2000

Eric Weisstein's World of Mathematics, Harmonic Mean

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A001008, A175441.

Sequence in context: A027271 A073991 A052642 * A081528 A056147 A181857

Adjacent sequences: A102925 A102926 A102927 * A102929 A102930 A102931

Keyword

nonn,frac

Author

Eric W. Weisstein, Jan 19 2005

Extensions

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

Status

approved