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

Author

Olivier Gérard

Keywords

• nonn

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

Examples

			(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.
		

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

Crossrefs

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

Programs

Magma

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

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 *)

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

Extensions

Corrected by Vladeta Jovovic, Aug 09 2002