3,1

Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,p) is divisible by p^3 for prime p>3. See A119722(n). The numerator of generalized harmonic number H(p-1,p^2) is divisible by p^4 for prime p>3. In general, the numerator of generalized harmonic number H(p-1,p^k) is divisible by p^(k+2) for prime p>3.

Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 29 2007, Table of n, a(n) for n = 3..6

Eric Weisstein, Link to a section of The World of Mathematics: Wolstenholme's Theorem.

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

a(n) = Numerator[ Sum[ 1/k^(Prime[n]^2), {k,1,Prime[n]-1} ] ] / Prime[n]^4 for n>2.

Prime[3] = 5.

a(3) = numerator[ 1 + 1/2^25 + 1/3^25 + 1/4^25 ] / 5^4 = 953962194872104906726451875/625 = 1526339511795367850762323.

Table[ Numerator[ Sum[ 1/k^(Prime[n]^2), {k, 1, Prime[n]-1} ] ] / Prime[n]^4, {n, 3, 10} ]

Cf. A119722 = Numerator of generalized harmonic number H(p-1, p)= Sum[ 1/k^p, {k, 1, p-1}] divided by p^3 for prime p>3.

Sequence in context: A003813 A003806 A095442 this_sequence A104304 A104322 A030198

Adjacent sequences: A130679 A130680 A130681 this_sequence A130683 A130684 A130685

frac,nonn,uned,bref

Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 29 2007