1,3

Up to n=18 a(n) is the same as A058313[n]=numerator of the n-th alternating harmonic number, sum ((-1)^(k+1)/k, k=1..n). a(n) differs from A058313[n] only for n=18, 28, 87, 99.

Up to n=100 the ratio a(n)/A058313[n]={1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1}.

A Wolstenholme-like theorem: for prime p > 3, if p = 6k-1, then p divides a(4k-1), otherwise if p = 6k+1, then p divides a(4k). Sum[Sum[(-1)^(i+j)*i/j, {i, 1, n}], {j, 1, n}] = -1/4(2(-1)^n*n+(-1)^n-1) * Sum[(-1)^(k+1)*1/k, {k, 1, n}].

a(n) = Abs[Numerator[Sum[Sum[(-1)^(i+j)*i/j,{i,1,n}],{j,1,n}]]].

Abs[Numerator[Table[Sum[Sum[(-1)^(i+j)*i/j, {i, 1, n}], {j, 1, n}], {n, 1, 50}]]]

Cf. A058313.

Sequence in context: A066219 A075830 A058313 this_sequence A119787 A025530 A106114

Adjacent sequences: A120298 A120299 A120300 this_sequence A120302 A120303 A120304

frac,nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 12 2006