1,2

a(n) almost always equals A082687(n)=Numerator[Sum[1/(n+k),{k,1,n}]], but differs at n=14,53,98,105,111,114,119,164,..

a(n) = Numerator[Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}]] - Numerator of the sum of all matrix elements of n X n Hilbert Matrix M(i,j) = 1/(i+j-1), (i,j=1..n). Denominator is A117664(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 23 2006

p divides a((p-1)/3) for prime p=7,13,19,31,37,43,61,67..=A002476[n] Primes of form 6n + 1. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 16 2006

Eric Weisstein's World of Mathematics, Harmonic Number

Eric Weisstein's World of Mathematics, Hilbert Matrix

a(n) = Numerator[n*Sum[1/(n+k),{k,1,n}]]. a(n) = Numerator[n*(Psi(2*n+1)-Psi(n+1))]. a(n) = Numerator[n*Sum[((-1)^(k+1)/k,{k,1,2n}] = Numerator[n*A058313(2n)/A058312(2n)].

n=2: HilbertMatrix[n,n]

1 1/2

1/2 1/3

so a(2) = Numerator[(1 + 1/2 + 1/2 + 1/3)] = Numerator[7/3] = 7.

The n X n Hilbert matrix begins:

1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...

1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...

1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...

1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...

1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...

1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...

Numerator[Table[n*Sum[1/(n+k), {k, 1, n}], {n, 1, 100}]]

Numerator[Table[Sum[Sum[1/(i+j-1), {i, 1, n}], {j, 1, n}], {n, 1, 30}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 23 2006

Cf. A082687, A058313, A058312.

Cf. A117664, A005249, A098118, A086881, A001008.

Cf. A002476.

Sequence in context: A093168 A097493 A082687 this_sequence A080174 A127729 A129736

Adjacent sequences: A117728 A117729 A117730 this_sequence A117732 A117733 A117734

frac,nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 14 2006