A130682 Numerator of generalized harmonic number H(p-1,p^2) = Sum_{k=1..p-1} 1/k^(p^2) divided by p^4 for prime p>3.

1526339511795367850762323, 187024220802620550798074497168768775337833066860651232788557036897081398718783708709

Offset: 3

Author

Alexander Adamchuk, Jun 29 2007

Keywords

• frac
• nonn
• uned
• bref

Comments

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

Examples

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

Links

• Alexander Adamchuk, Jun 29 2007, Table of n, a(n) for n = 3..6
• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

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.

Programs

Mathematica

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

Formula

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