A027459 Numerator of Sum_{k=1..n} H(k)/k, where H(k) is k-th harmonic number.

1, 7, 85, 415, 12019, 13489, 726301, 3144919, 30300391, 32160403, 4102360483, 4301068993, 758647585777, 112686856171, 3336876977, 96568406789, 28776062218037, 29608882035581, 1568274265798307, 11256448518043769

Offset

1,2

Comments

Originally defined as the first column of A027447, but now contains numerator in reduced form (cf. A329108). - Sean A. Irvine, Nov 04 2019

Numerators of the binomial transform of (-1)^n/(n+1)^3. The matrix a[i,j] below is the product of the binomial matrix and the matrix with general term binomial(i,j)(-1)^(i-j)/(i+1)^3. - Paul Barry, Aug 06 2004

From Alexander Adamchuk, Jan 02 2007 [edited by Jon E. Schoenfield, Mar 08 2015]: (Start)

Also a(n) is a numerator of S(n) = Sum_{k=1..n} H(k)/k, where H(k) is the k-th harmonic number, H(k) = Sum_{i=1..k} 1/i = A001008(k)/A002805(k).

S(n) = Sum_{k=1..n} H(k)/k = 1/2*(H(n)^2 + H(n,2)), where H(n,2) = Sum_{i=1..n} 1/i^2 = A007406(n)/A007407(n).

p divides a(p-1) and a(p-2) for prime p>3. a(n) is prime for n = {2, 7, 26, 31, 43, 53, 68, 80, 91, 123, 175, 236, 458, ...}. (End)

The n-fold repeated integral of (1/2)*log(x)^2 (all improper integrals with the lower limits of integration equal to 0) = x^n/n! * ( (1/2)*log(x)^2 - H(n)*log(x) + Sum_{k = 1..n} H(k)/k ). - Peter Bala, Feb 17 2022

Links

Alexander Adamchuk, Table of n, a(n) for n = 1..30

Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.

Eric Weisstein's World of Mathematics, Harmonic Number

Formula

Numerators of sequence a(1, n) in (a(i, j))^3 where a(i, j) = 1/i if j <= i, 0 if j > i.

Numerators of (Wolstenholme(n, 1)^2 + Wolstenholme(n, 2))/(2*n)= ((gamma+Psi(n+1))^2 + Pi^2/6 - Psi(1, n+1))/(2*n), where Wolstenholme(n, m) = Sum_{i=1..n} 1/i^m. - Vladeta Jovovic, Aug 09 2002

a(n) = numerator(Sum_{k=1..n} ((Sum_{i=1..k} 1/i)/k)). - Alexander Adamchuk, Jan 02 2007

Example

(a[ i,j ])^3 = MATRIX([[1, 0, 0, 0, 0], [7/8, 1/8, 0, 0, 0], [85/108, 19/108, 1/27, 0, 0], [415/576, 115/576, 37/576, 1/64, 0], [12019/18000, 3799/18000, 1489/18000, 61/2000, 1/125]]), n = 5.

Mathematica

Table[Numerator[Sum[Sum[1/i, {i, 1, k}]/k, {k, 1, n}]], {n, 1, 30}] (* Alexander Adamchuk, Jan 02 2007 *)
With[{nn=20}, Accumulate[HarmonicNumber[Range[nn]]/Range[nn]]]//Numerator (* Harvey P. Dale, Feb 26 2023 *)

Prog

(Magma) [Numerator(&+[HarmonicNumber(k)/k:k in [1..n]]):n in [1..20]]; // Marius A. Burtea, Nov 05 2019

Crossrefs

Cf. A001008, A002805, A007406, A007407, A027447, A329108.

Sequence in context: A183177 A367351 A058795 * A329108 A162160 A027531

Adjacent sequences: A027456 A027457 A027458 * A027460 A027461 A027462

Keyword

nonn

Author

Olivier Gérard

Extensions

Corrected by Vladeta Jovovic, Aug 09 2002

Status

approved