A125194 Numerator of generalized harmonic number H((p-1)/2,2p)= Sum[ 1/k^(2p), {k,1,(p-1)/2}] divided by p^2 for prime p>3.

41, 1599366601, 10877829357646990581304675244472669289, 100935935338172297894217692920950359818733561, 9217176064595104612826996436899733706027947436610177335077693637792069056822883934927465549747441

Offset: 3

Author

Alexander Adamchuk, Jan 13 2007

Keywords

• frac
• nonn

Comments

Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,2p) is divisible by p^2 for prime p>3 (see A120290(n)). The numerator of generalized harmonic number H((p-1)/2,2p) is divisible by p^2 for prime p>3.

Examples

			Prime[3] = 5.
a(3) = Numerator[ 1 + 1/2^10 ] / 5^2 = 1025 / 25 = 41.
		

Links

• Eric Weisstein's World of Mathematics, Harmonic Number
• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

Crossrefs

Cf. A120290, A119722, A001008, A007406, A007408, A007410.

Programs

Mathematica

Do[p=Prime[k];f=0;Do[f=f+1/n^(2p);g=Numerator[f];If[IntegerQ[g/(p)^2],Print[{p,g/p^2}]],{n,1,(p-1)/2}],{k,1,100}]

Formula

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