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A027459
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Numerator of Sum_{k=1..n} H(k)/k, where H(k) is k-th harmonic number.
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5
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1, 7, 85, 415, 12019, 13489, 726301, 3144919, 30300391, 32160403, 4102360483, 4301068993, 758647585777, 112686856171, 3336876977, 96568406789, 28776062218037, 29608882035581, 1568274265798307, 11256448518043769
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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(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.
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MATHEMATICA
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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 *)
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PROG
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(Magma) [Numerator(&+[HarmonicNumber(k)/k:k in [1..n]]):n in [1..20]]; // Marius A. Burtea, Nov 05 2019
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CROSSREFS
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Cf. A001008, A002805, A007406, A007407, A027447, A329108.
Sequence in context: A254569 A183177 A058795 * A329108 A162160 A027531
Adjacent sequences: A027456 A027457 A027458 * A027460 A027461 A027462
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KEYWORD
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nonn
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AUTHOR
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Olivier Gérard
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EXTENSIONS
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Corrected by Vladeta Jovovic, Aug 09 2002
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STATUS
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approved
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