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

%I #15 Feb 28 2023 07:43:55

%S 2479157521,159936660724017234488561,

%T 1119583852472161859174156302552583713828739479026834819554843860744244189

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

%C 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.

%H Alexander Adamchuk, <a href="/A120290/b120290.txt">First 5 terms</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>.

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

%e With prime(3) = 5, a(3) = numerator[ 1 + 1/2^10 + 1/3^10 + 1/4^10 ] / 5^2 = 61978938025 / 25 = 2479157521.

%t Table[Numerator[Sum[1/k^(2*Prime[n]),{k,1,Prime[n]-1}]],{n,3,7}]/Table[Prime[n]^2,{n,3,7}]

%Y Cf. A119722, A099828, A099827, A001008, A007406, A007408, A007410.

%K frac,nonn,bref

%O 3,1

%A _Alexander Adamchuk_, Jul 08 2006