A175441 Denominator of the harmonic mean of the first n positive integers.

1, 3, 11, 25, 137, 49, 363, 761, 7129, 7381, 83711, 86021, 1145993, 1171733, 1195757, 2436559, 42142223, 14274301, 275295799, 11167027, 18858053, 19093197, 444316699, 1347822955, 34052522467, 34395742267, 312536252003, 315404588903, 9227046511387, 9304682830147

Offset: 1

Author

Jaroslav Krizek, May 16 2010

Keywords

• nonn
• easy
• frac

Comments

See A102928 - numerators of the harmonic means of the first n positive integers.

a(n) = A001008(n) for n = 1 - 19 and other n.

a(n) is also the numerator of H(n)/(n+1)+1/(n+1)^2 = -int(x^n*log(1-x), x=0..1) with H(n) = A001008(x)/A002805(n) harmonic number of order n. - Groux Roland, Jan 08 2011

a(n) coincides with A001008(n) iff n is not in the sequence A256102. For the quotient A001008(n) / a(n) if n is from A256102 see the corresponding entry of A256103. - Wolfdieter Lang, Apr 23 2015

Examples

			H(n) = 1, 4/3, 18/11, 48/25, 300/137, 120/49, 980/363, 2240/761, ...
Comparison with A001008: the first 19 entries coincide because 20 is the first entry of A256102; indeed, A001008(20) = 55835135 and a(2) = 11167027. The quotient is 5 = A256103(1). - _Wolfdieter Lang_, Apr 23 2015
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A102928 (numerators), A001008, A256102, A256103.

Programs

Mathematica

Table[Denominator[HarmonicMean[Range[n]]],{n,30}] (* Harvey P. Dale, May 21 2021 *)

PARI

a(n)={denominator(n/sum(k=1, n, 1/k))} \\ Andrew Howroyd, Jan 08 2020

Formula

a(n) = denominator(n/(Sum_{k=1..n} 1/k)). - Andrew Howroyd, Jan 08 2020

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

Extensions

Terms a(25) and beyond from Andrew Howroyd, Jan 08 2020