A175441 Denominators of the harmonic means H(n) 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

1,2

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

Links

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

Formula

a(n) = numerator(sum(1/(k*(k+n)), k=1..oo)). - Paolo P. Lava, Jan 17 2013

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A190476 A060746 A111935 * A001008 A231606 A096617

Adjacent sequences: A175438 A175439 A175440 * A175442 A175443 A175444

Keyword

nonn,easy,frac

Author

Jaroslav Krizek, May 16 2010

Extensions

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

Status

approved