1,2

First column of A027447.

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 (pbarry(AT)wit.ie), Aug 06 2004

Also a(n) is a numerator of S(n) = Sum[ H(k)/k, {k,1,n} ], where H(k) is harmonic number, H(k) = HarmonicNumber[k] = Sum[ 1/i, {i,1,k} ] = A001008(k)/A002805(k). S(n) = Sum[ H(k)/k, {k,1,n} ] = 1/2*( H(n)^2 + H(n,2) ), where H(n,2) = HarmonicNumber[n,2] = Sum[ 1/i^2, {i,1,n} ] = 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,...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 02 2007

Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 02 2007, Table of n, a(n) for n = 1..30

Eric Weisstein, Link to a section of The World of Mathematics. Harmonic Number.

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 (vladeta(AT)eunet.rs), Aug 09 2002

a(n) = Numerator[ Sum[ Sum[ 1/i, {i,1,k} ] /k, {k,1,n} ] ]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 02 2007

(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.

Table[Numerator[Sum[Sum[1/i, {i, 1, k}]/k, {k, 1, n}]], {n, 1, 30}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 02 2007

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

Sequence in context: A166178 A034323 A058795 this_sequence A162160 A027531 A155459

Adjacent sequences: A027456 A027457 A027458 this_sequence A027460 A027461 A027462

nonn

Olivier Gerard (olivier.gerard(AT)gmail.com)

Corrected by Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 09 2002